The Importance of Intuition and Proof

A proof is, most essentially, a convincing explanation. It need not follow a rigid set of rules, and a proof is more than just right or wrong. Proofs can also be elegant or ugly. They can be clarifying or confusing. They can be illuminating or obscuring. And great proofs often break the rules, like this proof without words:

Can you prove the Pythagorean theorem with only pictures?

But at times, it's just not clear whether an explanation is right or wrong, especially in calculus, where we venture beyond the finite and graze hips with infinity, touching on a topic of which we have no tangible experience. Our intuition might even mislead us; for example, consider the following famous paradox of Aristotle—it seems convincing, but can't be right:

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. (Aristotle, Physics VI:9, 239b15)

At times like these, intuition isn't enough, and we formalize our proofs to demolish all doubt. In each area of math, we decide which facts can be assumed and which sorts of arguments are valid, and these standards govern our proofs. In our proofs, we leave out no detail, giving a reason for every statement and making sure our reasons are assumable or already-proven facts. The result is a body of knowledge, and its facts can only be questioned by questioning the decisions made at the outset.

So we want proofs that are both enlightening and precise, which creates a tension. Should we add a few lines of explanation to our proof without words to make it complete, or leave it as is to keep it concise? The answer is up for debate, and might depend on who our proof is designed to convince. If the proof is being written for a class, be sure to clarify how much detail your professor expects. However, if you are giving a talk, it may be wise to omit some of the nitty-gritty details but mention that there are more details the viewer can work on by him/herself.

But by asking this sort of question as we write our own proofs, weighing elegance against formality, we're likely to end up with a proof that truly convinces.

I owe the idea of a proof as a "convincing explanation" to Steven Strogatz; see this article, or his book, The Joy of X.