This course will most likely be very different from math courses you have taken previously. Chances are your mathematics education up to now has focused on computation. In high-school algebra you learned to solve linear and quadratic equations. In Calculus you learned to compute integrals and derivatives and how to use them to understand properties of functions. (In fact, Calculus is a Latin term that refers to a method for computation. In the past, what we study now as Calculus was referred to as “The Calculus of Infinitesimals” and later simply “The Calculus.” The term algebra is similarly derived from an Arabic word al jabr describing a step in solving linear equations.)
The focus of this course is very different. There will be some calculation — most importantly we will study a propostional calculus for combining mathematical statements. However, our main goal is something else entirely: you will learn how to formulate and write rigorous mathematical proofs.
Much of the activity of higher mathematics is focused on discovering and proving new mathematical results, or theorems. Computation plays an important role in this process, and you will continue to learn new computational methods as you proceed through your mathematical career. However, computation is only one step in completing work on a mathematical problem. In the end we need to communicate our work. And for this we must use language.
Mathematics is a human endeavor. Mathematical writing is first, and foremost, writing, i.e., symbolic communication between humans. Students often develop the mistaken impression that good mathematical writing is complicated and requires complicated and arcane symbols. The opposite is true. Good mathematical writing should be good writing. As such it should convey its content as simply as possible. As you write up your proofs, remember that each proof is first of all an essay in the English language. It may be a technical essay employing come specialized mathematical symbols and equations. However, all such technicalities should be bound up in a wrapper of explanatory language.
As you move on through a career in which you use your mathematics degree, you will repeatedly face the challenging task of conveying mathematical ideas to others. How you succeed will may be determined as much by your ability to explain as your ability to come up with new ideas. So it behooves you to practice this skill now.
There is another benefit to learning to write well. The act of writing forces you to reexamine something that you have understood in a different way. As you wrestle with putting your insights into words a greater level of understanding may emerge. In my own work, I have found that I do not truly understand something unless I know how to explain it to someone else.
Like computation, writing mathematics does not usually come naturally. It requires dedicated study and practice. This course is here to get you started on that study and practice. The textbook for the course has several chapters on writing and we will discuss these in detail in lecture and recitation.