I start by invoking the ‘fundamental rule of calculus notation’ to ensure the correct translation of an English sentence into the language of calculus. As examples, I derive the ‘rocket equation’ and the standard expression for the gravitational potential of a sphere. I discuss the importance of treating units properly. I make the important point that a frame is not the same as a system of coordinates. I distinguish between ‘proper vectors’ and ‘coordinates vectors’, which is needed for a proper understanding of transforming coordinates. Because the study of vehicle attitude is built on basis vectors, I show how to construct these from both an intuitive viewpoint and a purely mathematical viewpoint.

This paper discusses the place of the infinite in Kant’s philosophy, in particular as required for continuity in mathematics and physics. A fine-grained examination of the roles that the infinite and the infinitesimal play in Kant’s theory that illuminates the notion of construction in Kant’s philosophy of mathematics also uncovers challenges to certain prominent interpretations of Kant’s reliance on logic and intuition in mathematics.

This paper tracks the shift and the continuity in Kant’s views about the relation between mathematics and physics from the early precritical Physical Monadology (1756) up to the middle critical Metaphysical Foundations of Natural Science (1786) and compares the ways that Kant uses the mathematical ideas of infinite divisibility and the notion of infinitesimal to ground basic metaphysical notions such as contact and corporeal nature.