Foundations of Advanced Mathematics

The main reason, of course, is to be sure that something is true. Contrary to popular misconception, mathematics is not a formal game in which we derive theorems from arbitrarily chosen axioms. Rather, we discuss various types of mathematical objects, some geometric (for example, circles), some algebraic (for example,polynomials), some analytic (for example, derivatives) and the like. To understand these objects fully, we need to use both intuition and rigor. Our intuition tells us what is important, what we think might be true, what to try next and so forth. Unfortunately, mathematical objects are often so complicated or abstract that our intuition at times fails, even for the most experienced mathematicians. We use rigorous proofs to verify that a given statement that appears intuitively true is indeed true.

Another use of mathematical proofs is to explain why things are true, though not every proof does that. Some proofs tell us that certain statements are true, but shed no intuitive light on their subjects. Other proofs might help explain the ideas that underpin the result being proved; such proofs are preferable, though any proof, even if non-intuitive, is better than no proof at all.

A third reason for having proofs in mathematics is pedagogical. A student (or experienced mathematician for that matter) might feel that she understands a new concept, but it is often only when attempting to construct a proof using the concept that a more thorough understanding emerges.

Finally, a mathematical proof is a way of communicating to another person an idea that one person believes intuitively, but the other does not.

One final comment on writing proofs: neither thinking up proofs nor writing them properly is easy, especially as the material under consideration becomes more and more abstract. Mathematics is not a speed activity, and you should not expect to construct proofs rapidly. You will often need to do scratch work first, before writing up the actual proof. As part of the scratch work, it is very important to figure out the overall strategy for the problem being solved, prior to looking at the details. What type of proof is to be used? What definitions are involved? Not every choice of strategy ultimately works, of course, and so any approach needs to be understood as only one possible way to attempt to prove the theorem. If one approach fails, try another. Every mathematician has, in some situations, had to try many approaches to proving a theorem before finding one that works; the same is true for students of mathematics.