Physical Sciences and Mathematics Commons^™

Gödel's Theorem In The Continuing Education Of Mathematics Teachers, Ana J. Lemes

Journal of Humanistic Mathematics

The notion of dépaysement épistémologique (epistemological disorientation) aims to capture the sense of disorientation when a learner is led to question their prior assumptions and understandings, generating uncertainty in a context in which they thought they had certain knowledge. This article describes an activity used with a group of practicing mathematics teachers in Uruguay that integrates elements of the history of mathematics related to Gödel’s incompleteness theorem, with the aim of provoking in the participants the experience of dépaysement épistémologique. Results show that several of the teachers participating in the activity felt dépaysement épistémologique, and this feeling triggered …

The Nothing That Really Matters, Szilárd Svitek

Journal of Humanistic Mathematics

Zero has (a) special role(s) in mathematics. In the current century, we take negative numbers and zero for granted, but we should also be aware that their acceptance and their emergence in mathematics, and their ubiquity today, have not come to happen as rapidly as, for example, that of natural numbers. Students can quickly become confused by the question: is zero a natural number? The answer is simple: a matter of definition. The history of zero and that of negative numbers are closely linked. It was in the calculations of debts that the negative numbers first appeared, where the state …

Archimedes Of Syracuse And Sir Isaac Newton: On The Quadrature Of A Parabola, Wyatte C. Hooper

Journal of Humanistic Mathematics

Good mathematics stands the test of time. As culture changes, we often ask different questions, bringing new perspectives, but modern mathematics stands on ancient discoveries. Isaac Newton’s discovery of calculus (along with Leibniz) may seem old but is predated by Archimedes’ findings. Current mathematics students should be familiar with parabolas and simple curves; in our introductory calculus courses, we teach them to compute the areas under such curves. Our modern approach derives its roots from Newton’s work; however, we have filled in many of the gaps in the pursuit of mathematical rigor. What many students may not know is that …

“It’S All For The Best”: Optimization In The History Of Science, Judith V. Grabiner

Journal of Humanistic Mathematics

Many problems, from optics to economics, can be solved mathematically by finding the highest, the quickest, the shortest—the best of something. This has been true from antiquity to the present. Why did we start looking for such explanations, and how and why did we conclude that we could productively do so? In this article we explore these questions and tell a story about the history of optimization. Scientific examples we use to illustrate our story include problems from ancient optics, and more modern questions in optics and classical mechanics, drawing on ideas from Newton’s and Leibniz’s calculus and from the …

Mathematical Conquerors, Unguru Polarity, And The Task Of History, Mikhail Katz

Journal of Humanistic Mathematics

I compare several approaches to the history of mathematics recently proposed by Blåsjö, Fraser–Schroter, Fried, and others. I argue that tools from both mathematics and history are essential for a meaningful history of the discipline.

In an extension of the Unguru–Weil controversy over the concept of geometric algebra, Michael Fried presents a case against both Andr ́e Weil the “privileged observer” and Pierre de Fermat the “mathematical conqueror.” Here I analyze Fried’s version of Unguru’s alleged polarity between a historian’s and a mathematician’s history. I identify some axioms of Friedian historiographic ideology, and propose a thought experiment to gauge its …

Teaching History Of Mathematics: A Dialogue, Benjamin Braun, Eric Kahn

Journal of Humanistic Mathematics

Many colleges and universities offer a course in the history of mathematics. While the potential benefits for students taking such a course might be apparent, it is often less clear how teaching a history of mathematics course can be a transformational experience for faculty. We present a dialogue between the authors regarding their experiences teaching history of mathematics courses, including their motivation for doing so, the impact these experiences have had on their classroom practices and assessment methods, and the opportunities history of mathematics courses offer for incorporating social justice, equity, and inclusion into the study of mathematics. Our goal …

Ways Of Relating To The Mathematics Of The Past, Michael N. Fried

Journal of Humanistic Mathematics

Historians of mathematics, by definition, look at mathematics of the past. But mathematicians, too, often look at mathematics of the past; mathematicians of the past themselves often looked very closely at mathematics of their own past. Is their relationship to the past the same as that of the historians? Is every view of the past an historical view? Indeed, is every historical view historical in the same way? Or is it possible that there are different kinds of relationships to the mathematics of the past? This paper will suggest that there are in fact a variety of such relationships. It …

The Discipline Of History And The “Modern Consensus In The Historiography Of Mathematics”, Michael N. Fried

Journal of Humanistic Mathematics

Teachers and students of mathematics often view history of mathematics as just mathematics as they know it, but in another form. This view is based on a misunderstanding of the nature of history of mathematics and the kind of knowledge it attempts to acquire. Unfortunately, it can also lead to a deep sense of disappointment with the history of mathematics itself, and, ultimately, a misunderstanding of the historical nature of mathematics. This kind of misunderstanding and the disappointment following from it--both raised to the level of resentment--run through the paper "A Critique of the Modern Consensus in the Historiography of …