Time, physics, and the paradoxes of continuity

A recent article in this journal proposes a radical reformulation of classical and quantum dynamics based on a perceived deficiency in current definitions of time. The argument is incorrect but the errors highlight aspects of the foundations of mathematics and physics that are commonly confused and misunderstood. For this reason, the article provides an important and heuristic opportunity to reexamine the types of time and non-standard analysis. This paper will discuss the differences between physical time and experiential time and explain how an expanded system of real analysis containing infinitesimals can resolve the paradoxes of continuity without sacrificing the modern edifice of mathematical physics.