Leibniz, Calculus, and The Hyperreal Numbers

Abstract

Our ideas revolving around Calculus, Philosophy, Law, and Theology are often so clouded that we forget to acknowledge the people behind these ideas. Through this way of thinking, we forget to look at the foundations that took countless years and even lifetimes to construct out of what we believe to be nothingness. What if I were to say everything mentioned in the first sentence was revolutionized by a German mathematician, philosopher, and logician's name is Gottfried Wilhelm Leibniz. The first part of this paper will focus on outlining his contributions to the foundations and invention of Calculus, disagreements between him and Newton, Leibniz's notation for Calculus, and other works in other areas such as law, metaphysics, and theology. This paper does not cover details including birth, death, spouses, etc. as those take away from the goals of this paper. The second part focuses on Abraham Robertson's construction of the Hyperreal Numbers and their applications proving that Leibniz's intuition of infinitesimals and Calculus correct. Accurate recognition of one's work is critical in maintaining not only credibility over future pieces of work but also recognizing the accomplishments of one's work. Understanding Leibniz's work and the instrumental construction of Calculus and infinitesimals allows us to also focus on the foundations of our modern societies and trace where many of our common ideas and innovations stem from. Ultimately, by the end of this paper, one should have a better understanding of the impact Leibniz, infinitesimals, and the foundational understandings of Calculus.