Unofficial/Short Course Title: Analysis
Spring 2025, Deep Springs College, Prof. Brian Hill
The Mathematical Association of America review begins, “This is the best Calculus textbook ever written.” Need more be said? Anyway, this is the principal text for the course, but we will be using it mostly for its Parts I and II (its first eight chapters). This book’s Part III can stand alone as the basis of a standard course in derivative and integral calculus. However our topic is the foundation of calculus, not calculus itself, and we will only get to Chapter 9 in Part III, which introduces the derivative.
There is a wealth of material that logically fits in between standard high school mathematics and calculus. This material has been deemed too hard by most schools — both those with college prep tracks and with colleges themselves. It is simply skipped! Only those that continue on to the junior level as mathematics majors get to see it, and they see it in a course that is often titled “Introduction to Analysis” or *ldquo;Introduction to Mathematical Reasoning.” It is perfect for the Deep Springs curriculum insofar as it requires no prerequisites beyond very good high school mathematics, and yet it is both more advanced and provocative than what is generally found in lower-division math courses.
The material is called “mathematical analysis” or “real analysis” and it includes many theorems that are absolutely essential to calculus but are typically used as if they are obvious and without proof. What students usually move on to in their lower division courses, with the time freed up by skipping the foundations of calculus are all sorts of pragmatic results — like the chain rule as a method for simplifying derivatives, or integration by parts as a common trick for doing integrals, and even more mundane rules.
One might think that with only so much time to either obtain and apply practical results or to do mathematical proofs, that one should spend one’s time on pedestrian problems. Michael Spivak, renowned for textbooks in mathematics ranging from the introductory level to textbooks used by mathematics graduate students, disagrees: “calculus ought to be the place in which to expect, rather than avoid, the strengthening of insight with logic.” He goes on: “precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions.”
A course in the topics of mathematical analysis that undergird calculus prepares one to think rigorously, to understand real analysis the way mathematicians do, and to understand fundamental theorems that are beautiful and significant in their own right. As a treat we will even go beyond the confines of the real numbers and in our final unit introduce imaginary numbers (also known as complex numbers). This will allow us to conclude with the 1799 proof by Gauss of The Fundamental Theorem of Algebra.