On Examples, Counterexamples, and Proof by Example

This note is motivated by two seemingly unrelated items that have recently appeared in the Pi Mu Epsilon Journal (http://www.wpi.edu/~pme/journal.html). In one, the editor of the Journal wrote of "mathematical cranks" and how frustratingly difficult it can be sometimes to get would-be "circle-squarers" and "angle- trisectors" to see the errors in their "proofs." The first part of this paper illustrates--by way of example!--how futile is a "proof" that is attempted with one or more examples. The second part of the paper follows up on a recent article in the PME Journal, "How Economists Use Mathematics to Show Why Some People Work So Much For So Little," and shows that the author's result is a consequence of the special characteristics of the Cobb-Douglas utility function.