College of the Holy Cross Mathematics and Computer Science
College of the Holy Cross Mathematics and Computer Science
Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred), or jlittle@holycross.edu
Office Hours: MW 1-3pm, TR 11-12noon, F 9-10am, and by appointment
Course Homepage: http://math.holycross.edu/~little/Principles00/Prin.html
The early part of the history of a mathematical subject is often characterized by explosive growth of knowledge, as new methods and tools are introduced. Problems that were previously difficult or even impossible to solve become feasible. Frequently, though, methods that clearly "work" are really only partially understood. This was true especially of the development of the calculus by Newton, Leibniz, and their successors in the 1600's and 1700's. Extremely clever, but intuitive and uncritical uses of the rules for derivatives and integrals eventually led to some apparently nonsensical results.
In your study of calculus to date, you have approached the subject from much the same intuitive point of view that Newton and Leibniz employed. You have learned how to apply the techniques of calculus to solve many types of problems, but the courses you have taken have probably not spent much time addressing exactly why those methods work (in the cases they do work), or their limitations (the exact hypotheses needed to make them work).
In the education of each mathematician, as in the history of the subject itself, there comes a time when the intuitive treatment of calculus must give way to a deeper understanding of the theoretical foundations. For you as a student of mathematics, that time is now! In this class we will essentially repeat some of the historical developments of the 19th century and study the principles of analysis, the theory of calculus that emerged at that time. These steps were first taken by Cauchy, Riemann, Weierstrass and others whose collective work put calculus on a sound footing and gave complete proofs of the key results. (It is important to note that this was anything but a case of pursuing mathematical "rigor" for its own sake. The concern over the theory of calculus was motivated largely by a desire for reliability in the application of calculus to physics and other areas of science.)
The remaining two days will be devoted to in-class exams. See below for tentative dates. There is a more detailed day-by-day course schedule available on the course homepage. As always, it may become necessary to add, delete, or rearrange topics as we progress through the course.
Together with the Algebraic Structures/Linear Algebra sequence, this course forms part of the "basic training" (i.e. "boot camp") for mathematics majors. These intermediate level courses serve as bridges from the basic mathematics you have seen in high school and in the calculus courses to the more advanced courses you will normally take in your junior and senior years.
One of the major goals of this course is to build your skills in developing and writing mathematical proofs. Make no mistake, there will almost certainly be times when the "boot camp" analogy seems painfully apt -- many Mathematics majors find these courses to be among the most challenging they take at Holy Cross because the whole way we work may well seem unfamiliar and baffling at first:
At times, you may ask why we are doing this at all. The ultimate answer is that this "abstract" proof-oriented work is the way all mathematics is communicated and in a sense it is what most of advanced mathematics is really about. So you have to be prepared for it if you decide to continue! The concept of mathematical proof is the unique and distinctive feature of this branch of knowledge; I think it is no exaggeration to say that it is one of the crowning achievements of the human intellect. Even applied mathematicians (ones who work on problems directly inspired by problems in the real world) must develop new ideas to solve those problems, and then provide convincing evidence (proofs) that what they claim is true so that others can follow what they do.
Lest this sound too grim, be aware that you will have lots of chances to develop and practice these new skills, and I will always be willing to give you the benefit of my experience working with this kind of mathematics! Even if you find this difficult at first, persistence and openness to a different way of thinking will usually pay off in time.
To get you more directly involved in the subject matter of this course, several times during the semester, the class will break down into groups of 3 or 4 students for one or more days, and each group will work together for a portion of those class periods on a group discussion exercise. The exercises will be made up by me. I will be present and available for questions and other help during these periods. At the conclusion of some of these discussions, at times the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will be responsible for a write-up of solutions for the questions from each discussion day, and those will be graded and and returned with comments.
Some of the other meetings of the class will be structured as lectures when that seems appropriate.
The assignments for the course will consist of:
If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.