Physical Sciences and Mathematics Commons^™

What Is A Number?, Nicholas Radley

HON499 projects

This essay is, in essence, an attempt to make a case for mathematical platonism. That is to say, that we argue for the existence of mathematical objects independent of our perception of them. The essay includes a somewhat informal construction of number systems ranging from the natural numbers to the complex numbers.

Foundational Mathematical Beliefs And Ethics In Mathematical Practice And Education, Richard Spindler

Journal of Humanistic Mathematics

Foundational philosophical beliefs about mathematics in the mathematical community may have an unappreciated yet profound impact on ethics in mathematical practice and mathematics education, which also affects practice. A philosophical and historical basis of the dominant platonic and formalist views of mathematics are described and evaluated, after which an alternative evidence-based foundation for mathematical thought is outlined. The dualistic nature of the platonic view based on intuition is then compared to parallel historical developments of universalizing ethics in Western thought. These background ideas set the stage for a discussion of the impact of traditional mathematical beliefs on ethics in the …

Ethics And Mathematics – Some Observations Fifty Years Later, Gregor Nickel

Journal of Humanistic Mathematics

Almost exactly fifty years ago, Friedrich Kambartel, in his classic essay “Ethics and Mathematics,” did pioneering work in an intellectual environment that almost self-evidently assumed a strict separation of the two fields. In our first section we summarize and discuss that classical paper. The following two sections are devoted to complement and contrast Kambartel’s picture. In particular, the second section is devoted to ethical aspects of the indirect and direct mathematization of modern societies. The final section gives a short categorization of various philosophical positions with respect to the rationality of ethics and the mutual relation between ethics and mathematics.

Wittgenstein On Miscalculation And The Foundations Of Mathematics, Samuel J. Wheeler

Philosophy Faculty Publications

In Remarks on the Foundations of Mathematics, Wittgenstein notes that he has 'not yet made the role of miscalculating clear' and that 'the role of the proposition: "I must have miscalculated"...is really the key to an understanding of the "foundations" of mathematics.' In this paper, I hope to get clear on how this is the case. First, I will explain Wittgenstein's understanding of a 'foundation' for mathematics. Then, by showing how the proposition 'I must have miscalculated' differentiates mathematics from the physical sciences, we will see how this proposition is the key to understanding the foundations of mathematics.

An Evolutionary Approach To Crowdsourcing Mathematics Education, Spencer Ward

Honors College

By combining ideas from evolutionary biology, epistemology, and philosophy of mind, this thesis attempts to derive a new kind of crowdsourcing that could better leverage people’s collective creativity. Following a theory of knowledge presented by David Deutsch, it is argued that knowledge develops through evolutionary competition that organically emerges from a creative dialogue of trial and error. It is also argued that this model of knowledge satisfies the properties of Douglas Hofstadter’s strange loops, implying that self-reflection is a core feature of knowledge evolution. This mix of theories then is used to analyze several existing strategies of crowdsourcing and knowledge …

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]

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]

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.

All At One Point: The New Physics Of Italo Calvino And Jorge Luis Borges, Mark Thomas Rinaldi

Dissertations, Theses, and Capstone Projects

This work of comparative literary criticism focuses on the presence of mathematical and scientific concepts and imagery in the works of Italo Calvino and Jorge Luis Borges, beginning with an historical overview of scientific philosophy and an introduction to the most significant scientific concepts of the last several centuries, before shifting to deep, scientifically-driven analyses of numerous individual fictions, and finally concluding with a meditation on the unexpectedly fictive aspects of science and mathematics. The close readings of these authors' fictions are contextualized with thorough explanations of the potential literary implications of theories from physics, mathematics, neuroscience and chaos theory. …

La Vie De Blaise Pascal Et Son Heritage Mathématique Et Philosophique, Katherine Weeks

Honors Theses

This paper will discuss the life of Blaise Pascal, his philosophy and mathematics. We will first study the life of Pascal, by looking at his family, the success of his family, and how he was educated. We will then move onto study the history of the arithmetic triangle, which eventually becomes known as Pascal’s Triangle. Finally we will look at Pascal’s Pensées. Finally we will conclude that Pascal is not only known for Pascal’s Triangle but also his legacy in the Philosophic world. It becomes clear that Pascal would not have been the mathematician he was without the philosophy, and …

The Calculus Of Variations, Erin Whitney

Honors Theses

The Calculus of Variations is a highly applicable and advancing field. My thesis has only scraped the top of the applications and theoretical work that is possible within this branch of mathematics. To summarize, we began by exploring a general problem common to this field, finding the geodesic be-tween two given points. We then went on to define and explore terms and concepts needed to further delve into the subject matter. In Chapter 2, we examined a special set of smooth functions, inspired by the Calabi extremal metric, and used some general theory of convex functions in order to de-termine …

Book Review: Philosophy Of Science After Feminism By Janet Kourany, Gizem Karaali

Pomona Faculty Publications and Research

Janet Kourany’s book is a strange one: published by Oxford University Press (as a part of its Studies in Feminist Philosophy series), it is an academically oriented book, but reading it, you sense that this is not yet another theoretical monograph. For Kourany has her ax to grind, and more importantly she has a program to promote. The program is for philosophers of science and is motivated and encouraged by the amazing work done in the past few decades by feminist scientists and feminist scholars of science, technology, and society. In the following I will try to explain why I …

Prove It!, Kenny W. Moran

Journal of Humanistic Mathematics

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.

Creating A Warmer Environment For Women In The Mathematical Sciences And In Philosophy, Samantha Brennan, Rob Corless

Samantha Brennan

No abstract provided.

Objectivity, Information, And Maxwell's Demon, Steven Weinstein

Dartmouth Scholarship

This paper examines some common measures of complexity, structure, and information, with an eye toward understanding the extent to which complexity or information‐content may be regarded as objective properties of individual objects. A form of contextual objectivity is proposed which renders the measures objective, and which largely resolves the puzzle of Maxwell's Demon.

Abstracting Aristotle’S Philosophy Of Mathematics, John J. Cleary

Research Resources

In the history of science perhaps the most influential Aristotelian division was that

between mathematics and physics. From our modern perspective this seems like an unfortunate deviation from the Platonic unification of the two disciplines, which guided Kepler and Galileo towards the modern scientific revolution. By contrast, Aristotle’s sharp distinction between the disciplines seems to have led to a barren scholasticism in physics, together with an arid instrumentalism in Ptolemaic astronomy. On the positive side, however, astronomy was liberated from commonsense realism for the conceptual experiments of Aristarchus of Samos, whose heliocentric hypothesis was not adopted by later astronomers because …

The Deconstruction Of Mathematics: A Criticism Of Reuben Hersh's What Is Mathematics, Really? And The Humanist Philosophy Of Mathematics, David J. Stucki

Mathematics Faculty Scholarship

Mathematics, as an academic discipline, has stood for many years as the last bastion against a growing tide of intellectual relativism that has become all but ubiquitous. More recently, however, efforts have been made to "humanize" mathematics by advocating a social-constructivist approach to the philosophy of mathematics, both in practice and education. This paper is intended to serve as a critical response to one advocate of this approach, Reuben Hersh (What Is Mathematics, Really?, 1997), and in the process a defense of Platonism.

The Voice Of The Phi Sigma -- 1881 -- Volume 03, No. 15, Phi Sigma

The Voice of the Phi Sigma

This item is part of the Phi Sigma collection at the College Archives & Special Collections department of Columbia College Chicago. Contact archives@colum.edu for more information and to view the collection.