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.

The Literary relations between La Fontaine and the "Astrée" of Honoré d'Urfé : a dissertation presented to the faculty of the department of philosophy of the university of Pennsylvania in partial fulfilment of the requirements for the degree of doctor of philosophy / by Walther P. Fischer

Collection : Publications of the University of Pennsylvania ; n° 6

Teachers' professional learning within communities of mathematics-for-teaching practice

Ontario bansho is an emergent mathematics instructional strategy used by teachers working within communities of practice that has been deemed to have a transformational effect on teachers' professional learning of mathematics. This study sought to answer the following question: How does teachers' implementation of Ontario bansho within their communities of practice inform their professional learning process concerning mathematics-for-teaching? Two other key questions also guided the study: What processes support teachers' professional learning of content-for-teaching? What conditions support teachers' professional learning of content-for-teaching? The study followed an interpretive phenomenological approach to collect data using a purposive sampling of teachers as participants. The researcher conducted interviews and followed an interpretive approach to data analysis to investigate how teachers construct meaning and create interpretations through their social interactions. The study developed a model of professional learning made up of 3 processes, informing with resources, engaging with students, and visualizing and schematizing in which the participants engaged and 2 conditions, ownership and community that supported the 3 processes. The 3 processes occur in ways that are complex, recursive, nonpredictable, and contextual. This model provides a framework for facilitators and leaders to plan for effective, content-relevant professional learning by placing teachers, students, and their learning at the heart of professional learning.

The historical game-changes in the philosophy of devotion and caste as used and misused by the Bhagavad-gita

This thesis considers that the purport of the Bhagavadgita is to prioritize the philosophy of loving devotion to God (bhakti), not the propagation of color-coded-caste (varna system). The distinction between bhakti and caste becomes clear when one sees their effect on human life and on the society. Jnana and karma, two of the other polarities with which the Gita contends, finally support bhakti towards betterment, not deterioration, if done selflessly and with balance. Caste, however, is a totally different tension, which is always detrimental to the well-being of the person and the society. In the Gita, the devotees' mystical or emotional love of, God apprehends their ~ oneness with the Supreme God and with all beings, and transcends the pitiless segregation of the caste system, and opens the path of salvation to all irrespective of race, color, caste, class or gender in life. In spite of much opposition from orthodoxy, the bhakti movement spread allover India, and bhakti itself rose to the level of orthodoxy and has become the faith of millions of people especially of the south, and surprisingly, of even of those of the so called highest caste. And yet, caste still remains as an indelible mark of every Hindu, even after they change their religion. Although caste is less venomous now, it is still openly present in all walks of Indian life and shows up its ugly head at important moments such as marriage, elections for public office, admission to school or employment. True, bhakti is the antidote for. caste; but only real bhakti can remove caste completely, not mere lip-service to it. This thesis claims that bhakti is the deliberate major thrust of the teaching of the Gita while caste seems to be a contradiction of this thrust.

Investigating Formative Assessment: Exploring the Impact on the Self-Efficacy and Motivation of Mathematics Students

This is a study of the implementation and impact of formative assessment strategies on the motivation and self-efficacy of secondary school mathematics students. An explanatory sequential mixed methods design was implemented where quantitative and qualitative data were collected and analyzed sequentially in 2 different phases. The first phase involved quantitative data from student questionnaires and the second phase involved qualitative data from individual student and teacher interviews. The findings of the study suggest that formative assessment is implemented in practice in diverse ways and is a process where the strategies are interconnected. Teachers experience difficulty in incorporating peer and self-assessment and perceive a need for exemplars. Key factors described as influencing implementation include teaching philosophies, interpretation of ministry documents, teachers’ experiences, leadership in administration and department, teacher collaboration, misconceptions of teachers, and student understanding of formative assessment. Findings suggest that overall, formative assessment positively impacts student motivation and self-efficacy, because feedback is provided which offers encouragement and recognition by highlighting the progress that has been made and what steps need to be taken to improve. However, students are impacted differently with some considerations including how students perceive mistakes and if they fear judgement. Additionally, the impact of formative assessment is influenced by the connection between self-efficacy and motivation, namely how well a student is doing is a source of both concepts.

Harminics of evolution : the philosophy of individual life basedupon natural science, as taught by modern masters of the law

UANL

Nietzsche's physiological philosophy of history

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Embracing the Crisis in the Foundations of Mathematics

The crisis in the foundations of mathematics is a conceptual crisis. I suggest that we embrace the crisis and adopt a pluralist position towards foundations. There are many foundations in mathematics. However, ‘many foundations’ (for one building) is an oxymoron. Therefore, we shift vocabulary to say that mathematics, as one discipline, is composed of many different theories. This entails that there are no absolute mathematical truths, only truths within a theory. There is no unified, consistent ontology, only ontology within a theory.

Inputs for the incorporation of the UNESCO guidelines on ICT competency standards for teachers : the training of teachers of mathematics in Central America

Resumen tomado de la publicaci??n

School of Mathematics Computing Guide

Guide for computing in the School of Mathematics. Intended for new staff and PG students. Originally written by Anton Prowse from a number of earlier documents.