The Writings of Josiah Royce: A Critical Edition, 2019

A scholarly introduction to 'Sketch of the Infinitesimal Calculus,' an early manuscript from Josiah Royce, c. 1880, which articulates an early pragmatic theory of epistemology, phenomenology, protosemiotics, & a nascent metaphysics of absolute idealism within a post-Kantian context.

This syllabus is for a course I teach on Calculus to high school students. Its introduction is helpful for those who wish to better understand the significance of mathematics in a liberal arts education.

A summary of a good book.

2007

This is a brief overview of some turning points in the history of infinitesimals.

These notes are being written for an introductory honors calculus class, Math 1551, at LSU in the Fall of 2011. The approach is quite different from that of standard calculus texts. (In fact if I had to choose a subtitle for these notes, it would be 'An Anticalculus-text Book'.) We use natural, but occasionally unusual, definitions for basic concepts such as limits and tangents. We also avoid several stranger aspects of the universe of calculus texts, such as counterintuitive notions of what counts as 'local maximum' or obsessing over 'convex up/down', and stay with practice that is consistent with the way mathematicians actually work. For most topics we show how to work with the method first and then go deeper into proofs and finer points. We prove several results in sharper formulations than seen in calculus texts. Among drastic departures from the standard approaches we work with the extended real line R * = R∪{−∞, ∞}, and define limits in such a way that no special exceptions need to be made for limits involving ±∞. We follow a consistent strategy of using suprema and infima, which form a running theme through the historical development of the real line and calculus. An entire chapter is devoted to convexity. For various corrections and comments I thank Justin Katz.

This is an interesting book.

The Mathematical Gazette, 2018

The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels -as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the 17 th through the 20 th centuries. We will also present 'didactic observations' at relevant places in the historical account.