On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

Integration of teacher education into the Swiss higher education system

The integration of specific institutions for teacher education into the higher education system represents a milestone in the Swiss educational policy and has broad implications. This thesis explores organizational and institutional change resulting from this policy reform, and attempts to assess structural change in terms of differentiation and convergence within the system of higher education. Key issues that are dealt with are, on the one hand, the adoption of a research function by the newly conceptualized institutions of teacher education, and on the other, the positioning of the new institutions within the higher education system. Drawing on actor-centred approaches to differentiation, this dissertation discusses system-level specificities of tertiarized teacher education and asks how this affects institutional configurations and actor constellations. On the basis of qualitative and quantitative empirical data, a comparative analysis has been carried out including case studies of four universities of teacher education as well as multivariate regression analysis of micro-level data on students' educational choices. The study finds that the process of system integration and adaption to the research function by the various institutions have unfolded differently depending on the institutional setting and the specific actor constellations. The new institutions have clearly made a strong push to position themselves as a new institutional type and to find their identity beyond the traditional binary divide which assigns the universities of teacher education to the college sector. Potential conflicts have been identified in divergent cognitive normative orientations and perceptions of researchers, teacher educators, policy-makers, teachers, and students as to the mission and role of the new type of higher education institution. - L'intégration dans le système d'enseignement supérieur d'institutions qui ont pour tâche spécifique de former des enseignants peut être considérée comme un événement majeur dans la politique éducative suisse, qui se trouve avoir des conséquences importantes à plusieurs niveaux. Cette thèse explore les changements organisationnels et institutionnels résultant de cette réforme politique, et elle se propose d'évaluer en termes de différentiation et de convergence les changements structurels intervenus dans le système d'éducation tertiaire. Les principaux aspects traités sont d'une part la nouvelle mission de recherche attribuée à ces institutions de formation pédagogique, et de l'autre la place par rapport aux autres institutions du système d'éducation tertiaire. Recourant à une approche centrée sur les acteurs pour étudier les processus de différen-tiation, la thèse met en lumière et en discussion les spécificités inhérentes au système tertiaire au sein duquel se joue la formation des enseignants nouvellement conçue et soulève la question des effets de cette nouvelle façon de former les enseignants sur les configurations institutionnelles et les constellations d'acteurs. Une analyse comparative a été réalisée sur la base de données qualitatives et quantitatives issues de quatre études de cas de hautes écoles pédagogiques et d'analyses de régression multiple de données de niveau micro concernant les choix de carrière des étudiants. Les résultats montrent à quel point le processus d'intégration dans le système et la nouvelle mission de recherche peuvent apparaître de manière différente selon le cadre institutionnel d'une école et la constellation spécifique des acteurs influents. A pu clairement être observée une forte aspiration des hautes écoles pédagogiques à se créer une identité au-delà de la structure binaire du système qui assigne la formation des enseignants au secteur des hautes écoles spéciali-sées. Des divergences apparaissent dans les conceptions et perceptions cognitives et normatives des cher-cheurs, formateurs, politiciens, enseignants et étudiants quant à la mission et au rôle de ce nouveau type de haute école. - Die Integration spezieller Institutionen für die Lehrerbildung ins Hochschulsystem stellt einen bedeutsamen Schritt mit weitreichenden Folgen in der Entwicklung des schweizerischen Bildungswesens dar. Diese Dissertation untersucht die mit der Neuerung verbundenen Veränderungen auf organisatorischer und institutioneller Ebene und versucht, die strukturelle Entwicklung unter den Gesichtspunkten von Differenzierung und Konvergenz innerhalb des tertiären Bildungssystems einzuordnen. Zentrale Themen sind dabei zum einen die Einführung von Forschung und Entwicklung als zusätzlichem Leistungsauftrag in der Lehrerbildung und zum andern die Positionierung der pädagogischen Hochschulen innerhalb des Hochschulsystems. Anhand akteurzentrierter Ansätze zur Differenzierung werden die Besonderheiten einer tertiarisierten Lehrerbildung hinsichtlich der Systemebenen diskutiert und Antworten auf die Frage gesucht, wie die Reform die institutionellen Konfigurationen und die Akteurkonstellationen beeinflusst. Auf der Grundlage qualitativer und quantitativer Daten wurde eine vergleichende Analyse durchgeführt, welche Fallstudien zu vier pädagogischen Hochschulen umfasst sowie Regressionsanalysen von Mikrodaten zur Studienwahl von Maturanden. Die Ergebnisse machen deutlich, dass sich der Prozess der Systemintegration und die Einführung von Forschung in die Lehrerbildung in Abhängigkeit von institutionellen Ordnungen und der jeweiligen Akteurkonstellation unterschiedlich gestalten. Es lässt sich bei den neu gegründeten pädagogischen Hochschulen ein starkes Bestreben feststellen, sich als neuen Hochschultypus zu positionieren und sich eine Identität zu schaffen jenseits der herkömmlichen binären Struktur, welche die pädagogischen Hochschulen dem Fachhochschul-Sektor zuordnet. Potentielle Konflikte zeichnen sich ab in den divergierenden kognitiven und normativen Orientierungen und Wahrnehmungen von Forschern, Ausbildern, Bildungspolitikern, Lehrern und Studierenden hinsichtlich des Auftrags und der Rolle dieses neuen Typs Hochschule.