Mathematical warrants, objects and actions in higher school mathematics

'Higher school mathematics' connotes typical upper secondary school and early college mathematics. The mathematics at this level is characterised by moves to (1) rigour in justification,(2) abstraction in content and (3) fluency in symbolic manipulation. This thesis investigates these three transitions - towards rigour, abstraction, and tluencyusing philosophical method: for each of the three transitions a proposition is presented and arguments are given in favour of that proposition. These arguments employ concepts and results from contemporary English language-medium philosophy and also rely crucially on classroom issues or accounts of mathematical experience both to elucidate meaning and for the domain of application. These three propositions, with their arguments, are the three sub-theses at the centre of the thesis as a whole. The first of these sub-theses (1) argues that logical deduction, quasi-empiricism and visualisation are mathematical warrants, while authoritatively based justification is essentially non-mathematical. The second sub-thesis (2) argues that the reality of mathematical entities of the sort encountered in the higher school mathematics curriculum is actual not metaphoric. The third sub-thesis (3) claims that certain 'mathematical action' can be construed as non-propositional mathematical knowledge. The application of these general propositions to mathematics in education yields the following: 'coming to know mathematics' involves:(1) using mathematical warrants for justification and self conviction; (2) ontological commitment to mathematical objects; and (3)developing a capability to execute some mathematical procedures automatically

The art and architecture of mathematics education: a study in metaphors

This chapter presents the summary of a talk given at the Eighth European Summer University, held in Oslo in 2018. It attempts to show how art, literature, and history, can paint images of mathematics that are not only useful but relevant to learners as they can support their personal development as well as their appreciation of mathematics as a discipline. To achieve this goal, several metaphors about and of mathematics are explored

THE CASE FOR AMERICAN HISTORY IN THE LAW SCHOOL CURRICULUM

This article argues for the teaching of American History throughout the first year of law school. I do not believe that students can fully understand the cases they are reading in other courses without a knowledge of environing context. Understanding American History -- which is many respects doesn\u27t paint a flattering picture -- may also help students in making fundamental choices about what role they wish to play in their careers as lawyers. I believe it is time to recognize that too much of the profession is run as a business and not as a noble calling dedicated to helping those who cannot help themselves. My own view is that the latter approach leads to happiness and peace of mind in the lives of lawyers and thus may counter the malaise that seems to have settled over the profession -- one in which attorneys are dissatisfied with their personal and professional lives

Pascal's wager

Pascal's Wager, discussed in his Pensées, has provoked discussion and strong views ever since its publication. In it, he proposes: Either God is or he is not. But to which view shall we be inclined? Reason cannot decide this question.ⁱ In this thesis I hope to make a contribution to the ongoing debate by setting Pascal's Wager into a modern decision-making context, providing a taxonomy of objections to the Wager and developing a critical framework which can be used to systematically examine each category in turn to see whether an objection holds. I will also present a new approach to handling 'mixed' strategies, as suggested by Alan Hájek and others, which uses a heuristic model of our perception of infinite rewards. I hold that Pascal's remedy for the unbeliever is a therapeutic response which is entirely in line with modern psychological practice and should not offend moral sensitivities, because it is purely an experiment to see whether faith can naturally arise once the objections are temporarily set aside. I argue that Pascal's Wager needs to be seen as an exercise in personal risk management and that Pascal anticipated both modern decision theory and the associated psychology of how we make choices in formulating his Wager. I suggest that if we understand it in this light, employing the critical toolkit that I assemble, then Pascal's Wager holds against all current objections. ⁱ Blaise Pascal, Pensées, trans. A. J. Krailsheimer (London: Penguin, 1995). 122. L41

Report of the Secretary of War; being part of the message and documents communicated to the two Houses of Congress at the beginning of the second session of the Fifty-fourth Congress, 1896

Annual Report of the Sec. of War. 24 Nov. HD 2, 54-2, v2-9, 5975p. (3478-3485] Pursuit of renegade Apaches; deportation of Canadian Crees; etc

Interpretation, the subject and the literature of Georges Bataille

This thesis pursues two closely related lines of argument. In the first half, I explore the Bataillean notion of man through his complex relationship with Hegel and Nietzsche. The Janus-like conception that will be dis-covered results from Bataille’s unwillingness to grant priority either to Hegel’s insights concerning the structure of consciousness or to Nietzsche’s claim, contra Hegel, that those putative insights ‘involve a vast and thorough corruption, falsification, superficialization, and generalization’ (The Gay Science) Bataille acknowledges the heuristic value of both thinkers’ work but ultimately refuses to let either become the dominant force within his thought. In the end Bataille’s human being remains caught between the ‘ex-cess’ of Nietzschean Will and the ‘restriction’ of Hegelian consciousness. He sees human existence, much like Freud, as moving with a ‘vacillating rhythm’ (Beyond the Pleasure Principle) between ‘conscious’ and ‘unconscious’ activity. This recognition leads him to conclude that there exists a fundamental ambiguity to human existence – the Impossible – which resists reduction or assimilation to any kind of formal discourse. The second half proceeds to explore this ambiguity in more detail by first teasing out the relationship be-tween the traumatic experiences at the heart of two of Bataille’s novels against the Freudian notion of Trauma (repetition automatism) and its relation to the creation of Identity. This ultimately proves insuffi-cient when it comes to interpreting the actions of Bataille’s fictional characters. However it opens a space within which other methodologies of interpretation, namely those of Lacan, Girard, and Derrida, can be in-vestigated as potential sources of insight into those characters’ psychological structures and motivation. Here they are explored in relation to each other and in order to describe and explain more adequately the ‘impossible’ ambiguity at the heart of Bataille’s novels and conception of the human