October 30, 1998

Proof

I tutor math, all levels of math, from basic algebra to complex proofs. Finding ways to teach is often a challenge, and the best way is to use analogies. I thought of a good one yesterday. Proof is a very difficult concept, for it involves not only knowing where you start from, but also where you're going. You see, proof is like driving a car down the road.

With the basic type of proof, you leave Chicago with some hypothesis in mind and begin cruising around in the hopes of eventually reaching your conclusion in St. Louis. Now, you just can't drive anywhere; you have to follow roads and highways. These are the rules of logic and definition. You must abide by these.

Now, you hope with each turn, off-ramp, and toll booth that you get closer and closer to St. Louis. If you reach St. Louis without a hitch, you have a proof.

If you meet a fiery demise in a car accident, it's not that your proof is wrong. A wrong proof will leave you stranded Des Moines or some other place that isn't quite St. Louis. Or, you might just drive around in circles forever.

A wreck, however, is a disproof, a counterexample. From where you began, you were never meant to reach your destination. There is no line of logic from your hypothesis to your conclusion, nor is there a highway from Denver to Casablanca.

Proof by contradiction, where you assume a theorem false and then try to find some illogical flaw that concludes from it, is like driving in reverse down the wrong way of the street. You're inviting a car accident. You want to die in a melting inferno. Now, you just might make it to St. Louis, so perhaps you actually have a normal proof here. Or, you might just need to alter those contradictions of yours and make them a wee bit stranger. Perhaps you should drive from the back seat as well while drunk.

Comments:

• =D you math boys crack me up!
  -Jeff