Logic and Foundations of Mathematics Commons^™

How Does “Collaboration” Occur At All? Remarks On Epistemological Issues Related To Understanding / Working With ‘The Other’, Don Faust, Judith Puncochar

Conference Presentations

Collaboration, if to occur successfully at all, needs to be based on careful representation and communication of each stakeholder’s knowledge. In this paper, we investigate, from a foundational logical and epistemological point of view, how such representation and communication can be accomplished. What we tentatively conclude, based on a careful delineation of the logical technicalities necessarily involved in such representation and communication, is that a complete representation is not possible. This inference, if correct, is of course rather discouraging with regard to what we can hope to achieve in the knowledge representations that we bring to our collaborations. We suggest …

Introduction To Functions And Generality Of Logic. Reflections On Frege's And Dedekind's Logicisms, Hourya Benis Sinaceur, Marco Panza, Gabriel Sandu

MPP Published Research

This book examines three connected aspects of Frege’s logicism: the differences between Dedekind’s and Frege’s interpretation of the term ‘logic’ and related terms and reflects on Frege’s notion of function, comparing its understanding and the role it played in Frege’s and Lagrange’s foundational programs. It concludes with an examination of the notion of arbitrary function, taking into account Frege’s, Ramsey’s and Russell’s view on the subject. Composed of three chapters, this book sheds light on important aspects of Dedekind’s and Frege’s logicisms. The first chapter explains how, although he shares Frege’s aim at substituting logical standards of rigor to intuitive …

Newton On Indivisibles, Antoni Malet, Marco Panza

MPP Published Research

Though Wallis’s Arithmetica infinitorum was one of Newton’s major sources of inspiration during the first years of his mathematical education, indivisibles were not a central feature of his mathematical production.

Wallis On Indivisibles, Antoni Malet, Marco Panza

MPP Published Research

The present chapter is devoted, first, to discuss in detail the structure and results of Wallis’s major and most influential mathematical work, the Arithmetica Infinitorum (Wallis 1656). Next we will revise Wallis’s views on indivisibles as articulated in his answer to Hobbes’s criticism in the early 1670s. Finally, we will turn to his discussion of the proper way to understand the angle of contingence in the first half of the 1680s. As we shall see, there are marked differences in the status that indivisibles seem to enjoy in Wallis’s thought along his mathematical career. These differences correlate with the changing …

Book Review: How To Bake Pi: An Edible Exploration Of The Mathematics Of Mathematics, Darren B. Glass

Math Faculty Publications

If you think about it, mathematics is really just one big analogy. For one example, the very concept of the number three is an drawing an analogy between a pile with three rocks, a collection of three books, and a plate with three carrots on it. For another, the idea of a group is drawing an analogy between adding real numbers, multiplying matrices, and many other mathematical structures. So much of what we do as mathematicians involves abstracting concrete things, and what is abstraction other than a big analogy? [excerpt]

The Influence Of Peer Educators In A First-Year Seminar On Freshman Preparation For Collegiate Challenges: A Qualitative Investigation Of Observational Learning, Sarah A. Forbes

All-Inclusive List of Electronic Theses and Dissertations

Most high school students have not spent deliberate time preparing for their transition to college. Knowing this, institutions have developed a first-year seminar geared toward transitional issues inherent to a specific institution. While the research on these programs illustrates their utility, there appears to be an opportunity to further their success by incorporating peers as educators in the classroom. Bandura (1986) saw the potential of observational learning through peer modeling, though few researchers have studied first-year seminars from this theoretical perspective. Through a postpositivistic philosophical paradigm, this exploratory qualitative study utilized a phenomenological design to investigate two research questions: what …

The Performance Of Leadership: All The School’S A Stage And The Teachers And Students Are Merely Players, Christi Anne Fenton

All-Inclusive List of Electronic Theses and Dissertations

The purpose of this qualitative case study was to identify and better understand the performance qualities that are prevalent in successful principals. Specifically, acting and leadership tools previously identified and compared by Dunklee (2000) were used to gain a better understanding of the interview process utilized to hire K-12 principals in school districts with more than 10,000 students in the state of Indiana. Arguably, the roles of the principal are much like those of an actor who must think on his or her feet in multiple situations. It is hoped that this study contributes new knowledge to assist in the …

Effective Practices For Student Success In Algebra, Michelle M. Ginkins

All-Inclusive List of Electronic Theses and Dissertations

The purpose of this quantitative study was to better understand the instructional strategies of Algebra I teachers and leadership characteristics of administrators among secondary public schools with high and low performance on Algebra I End-of-Course Assessments (ECAs). The impact of poverty on student achievement was also investigated in relation to first-time test taker Algebra 1 ECA passing rates. The study south to determine if schools’ free and reduced lunch rates, teacher characteristics, and/or principal characteristics were significant predictors of first-time test taker Algebra 1 ECA passing rates. Whether there were significant differences in implementation of Marzano, Pickering, and Pollock’s (2001) …

The Ecology, Biogeochemistry And Phylogeny Of Methane Seep And Non-Seep Benthic Foraminifera From The Pacific Margin, Ashley M. Burkett

All-Inclusive List of Electronic Theses and Dissertations

In an effort to understand the relationships between active methane seep and adjacent non-seep (inactive) populations of the deep-sea foraminifera Cibicidoides wuellerstorfi, a common paleo-indicator species, from methane seeps in the Pacific were analyzed and compared to one another for genetic similarities of small subunit rDNA (SSU rDNA) sequences. Pacific Ocean C. wuellerstorfi were also compared to those collected from other localities around the world (based on 18S gene available on Genbank, e.g., Schweizer et al., 2009). Results from this study revealed that C. wuellerstorfi living in seeps near Costa Rica and Hydrate Ridge are genetically similar to one another …

The Square Of Opposition: Innovations In Teaching Logic, Marc R. Dimartino

Senior Honors Projects

Teaching classical logic can often be challenging, especially when working with students who lack any prior experience with the more technical aspects of critical thinking. The abstraction of statements into logical symbols and the implementation of various diagramming methods can be enough to frustrate novice logicians, leading to a lack of hope and sometimes failure of mastery. The unique difficulties in teaching classical logic can, in addition, exacerbate tricky pedagogical issues that arise on a day to day basis in the critical thinking classroom. For example, it can be challenging to convey complex information in a meaningful way when dealing …

Thriving, Not Just Surviving, On Food Stamps, Sam L. Mccaughey

Senior Honors Projects

Food insecurity has been a persistent element in the history of the United States. Efforts to address the problem - and the larger issue of poverty - have been wide-ranging, but the debate about how best to respond to hunger has often centered on the relative roles of government and charity. Often that debate has led to hybrid solutions that combine government sponsored welfare such as food stamps and community-based food relief programs such as food banks. Yet, even such complementary approaches leave many people’s needs unmet, and there remains a significant population, both across the country and in Rhode …

A Qualitative Historiographical Case Study Of The Educative Properties Of Eugene V. Debs And John Dewey: 21st Century Implications For Education, Bradley Countermine

All-Inclusive List of Electronic Theses and Dissertations

This study aimed to shed light on the current state of educational reform rhetoric through an analysis of previous attempts to shape public education for the benefit of all. Analyzing Eugene V. Debs’s and John Dewey’s views on democracy and education during the Progressive Era promotes a version and vision of education that inspires people to think critically, to navigate contemporary society, and to acknowledge current issues within public education and United States society at large. Because education both reflects society and has the power to transform it, the struggle for fair, equitable, and enlightening education is paramount to the …

A Formalization Of Topical Logic, Aharon Grenadir

School for Lifelong Education Publications

The author discusses the history of topical logic.

On Pi Day, A Serving Of Why We Need Math, Darren B. Glass

Math Faculty Publications

Today, our Facebook feeds will be peppered with references to Pi Day, a day of celebration that has long been acknowledged by math fans and that Congress recognized in 2009. Every high schooler learns that pi is the ratio of the circumference of a circle to its diameter and that its decimal expansion begins 3.14 and goes on infinitely without repeating. [excerpt]

An Argument For A Neutral Free Logic, Daniel Yeakel

Wayne State University Dissertations

I argue for a neutral free logic is a logic wherein sentences containing non-referring terms do not have truth value. The primary support for this conclusion comes by way of criticism of the alternatives. If every sentence of the form `a = a' is a logical truth and is consequently knowable a priori then it will follow absurdly that `a exists' is knowable a priori. There are several alternatives for avoiding this intolerable conclusion and I argue that, with the exception of neutral free logic which holds that `a = a' can lack truth value, their successes are not sufficient …

A Reply To Heathcote's "On The Exhaustion Of Mathematical Entities By Structures", Teresa Kouri

Philosophy Faculty Publications

In this article I respond to Heathcote’s "On the Exhaustion of Mathematical Entities by Structures". I show that his ontic exhaustion issue is not a problem for ante rem structuralists. First, I show that it is unlikely that mathematical objects can occur across structures. Second, I show that the properties that Heathcote suggests are underdetermined by structuralism are not so underdetermined. Finally, I suggest that even if Heathcote’s ontic exhaustion issue if thought of as a problem of reference, the structuralist has a readily available solution.

The Logical System Of Frege’S Grundgesetze : A Rational Reconstruction, Méven Cadet, Marco Panza

MPP Published Research

This paper aims at clarifying the nature of Frege's system of logic, as presented in the first volume of the Grundgesetze . We undertake a rational reconstruction of this system, by distinguishing its propositional and predicate fragments. This allows us to emphasise the differences and similarities between this system and a modern system of classical second-order logic.

Pruebas Entimemáticas Y Pruebas Canónicas En La Geometría Plana De Euclides, Marco Panza, Abel Lassalle Casanave

MPP Published Research

Dado que la aplicación del Postulado I.2 no es uniforme en Elementos, ¿de qué manera debería ser aplicado en la geometría plana de Euclides? Además de legitimar la pregunta misma desde la perspectiva de una filosofía de la práctica matemática, nos proponemos esbozar una perspectiva general de análisis conceptual de textos matemáticos que involucra una noción ampliada de la teoría matemática como sistema de autorizaciones o potestades y una noción de prueba que depende del auditorio.

Since the application of Postulate I.2 in the Elements is not uniform, one could wonder in what way should it be applied in Euclid’s …

Denying Antecedents And Affirming Consequents: The State Of The Art, David Godden, Frank Zenker

Philosophy Faculty Publications

Recent work on conditional reasoning argues that denying the antecedent [DA] and affirming the consequent [AC] are defeasible but cogent patterns of argument, either because they are effective, rational, albeit heuristic applications of Bayesian probability, or because they are licensed by the principle of total evidence. Against this, we show that on any prevailing interpretation of indicative conditionals the premises of DA and AC arguments do not license their conclusions without additional assumptions. The cogency of DA and AC inferences rather depends on contingent factors extrinsic to, and independent of, what is asserted by DA and AC arguments. © David …

Philosophy Of Mathematics: Theories And Defense, Amy E. Maffit

Williams Honors College, Honors Research Projects

In this paper I discuss six philosophical theories of mathematics including logicism, intuitionism, formalism, platonism, structuralism, and moderate realism. I also discuss problems that arise within these theories and attempts to solve them. Finally, I attempt to harmonize the best features of moderate realism and structuralism, presenting a theory that I take to best describe current mathematical practice.