A Brief Introduction to Proofs

Proofs are perhaps the very heart of mathematics. Unlike the other sciences, mathematics adds a final step to the familiar scientific method. After experimenting, collecting data, creating a hypothesis, and checking that hypothesis through more experiments, mathematicians must prove their hypothesis is correct before it can be accepted as a true result. Similarly, mathematicians will only consider a piece of written mathematics a “paper” if it contains a proof. Otherwise it can be nothing more than an extended abstract at best. At the same time, students find reading and writing proofs among the most difficult topics in mathematics. Writing proofs, in particular, takes years of practice. Textbooks usually offer very little guidance, although there are a few books on the topic, such as Solow [2005]. However, at over 250 pages, it is a bit long to expect students to read on their own during a busy semester. I expect this short guide to proofs to serve as an introduction that students can reference throughout the semester. It will not replace longer reference guides such as Solow [2005], which students may still find useful.