The main goal of this course is to introduce you to the mathematical way of thinking and writing, particularly as this relates to proofs. In this course, you should learn to:
The content of the course will center around the study of:
- Write clear mathematical arguments,
- Think mathematically,
- Appreciate how writing and thinking depend on each other, and
- Understand the roles of:
- Precision, and
- Sustained effort
- Basic set theory
- Rational and irrational numbers
- Mathematical induction
- Equivalence relations and partitions
- Sizes of infinity
The course will include a considerable amount of writing, and thinking about writing. You will find this quite different from past mathematics courses you have taken, and you may find it frustrating; that is natural. This course is intended to help you bridge the gap between performing mathematical algorithms by rote memorization and actually doing mathematics yourself. This is a major change in thinking for many people, and it does not come easily.
One important thing to recognize is that you will not know how to do a problem when you first look at it. It frequently will take several sittings to come to a solution to a problem, and this may be contrary to your experience in math classes to date. Expect this to happen; it is the way that most real mathematical thinking occurs, and it is one of the things we are going to practice in this course. Uncertainty is not a bad thing; the methods of transforming the unknown into something understood are central to the study of mathematics, and will be a guiding theme for this class.
See also the course outline for the timing of these topics.