I was reading through Topology Now! by Robert Messer and Philip Straffin, a really good book with a unique expository structure which I have read twice, when I came across a proof that there exists “a real number whose square is .” The demonstration was in the text to show an application of the Completeness Property of Real Numbers. At one point in the proof is chosen such that . As is common in proofs, the expression seems to come out of the blue. By this I mean that, as becomes clear three lines later, the expression for is exactly the value that makes an inequality true and leads to a contradiction. But there is no logical process I can see by which the bound on could have been discovered before making a trial of those three steps. The must have been found in the process of developing/ discovering the proof and did not form out of thin air as it seems in the formal proof.
So there does exist a real number whose square is . We have a proof – a formal argument consisting of eliminating cases using proof by contradiction. We can check the proof for hidden assumptions, algebra errors, missing cases etc. If everything is okay, we have a new fact for our later use. But what have we learned? It depends on the proof. There may be obvious generalizations or a new technique to put in our tool box or as in the example at hand some insight into a mathematical structure. But do we really understand why is a real number?
One can imagine “teaching” the proof – emphasizing the crucial assumptions, outlining the structure, anticipating small difficulties. This would be an exercise in modeling how to read a proof. Or we can ask “Why is the statement of the theorem true?” If we had a model in our heads that helps us understand why, maybe we can extend the statement to say or . The model will be individual. Each of us has different neuronal models for mathematical concepts that we use for our own understanding and which we must have before we can explain ourselves to anyone else much less prove something. For the idea that the square root of exists, we might look at the length of the hypotenuse of an isosceles right triangle or consider a series of rational approximations to or think about the “empty” space on the number line. Building these models, that is, conceptual understanding of mathematical structures is much harder than understanding the steps of a proof. This is why we have math instructors and math courses.
Getting back to Topology Now! The particular proof I happened to read was intended to illustrate the power and necessity for The Completeness Property. It serves its purpose. This post is not a critique. I just happened to read it and my mind took off in this flight of fancy.