I had an amazing math teacher in college who taught me some important life lessons.
Among other things, he would often remind us that “A proof is nothing more than a convincing argument.”
This statement sounds obvious on the surface. But when buried in axioms and theorems and corollaries, it can be easy to miss.
Instead of intimidating us with rigor, this teacher would get us to slow down and settle down.
“Forget about the formalities for a second, what are we trying to show here? How do we know it? Let’s play at it before we work at it.”
One time we were working on a combinatorial problem. I didn’t know the proper way to solve it, but I noticed that its sample data set was tiny. This made it easy to list all the possibilities by rote, and then count up the ones that fit the rule by hand.
I did that, and blurted out the answer before anyone else had a chance to think. The professor asked me to convince him, and that forced me to reveal my not-so-clever trick.
His response was “I’m convinced. For this case. And that’s called proof by exhaustion — it’s a real technique! But as the name implies, it can get pretty exhausting with larger sets.”
He added an element to the set. Then another, then another, until his point was crystal clear.
At first I was thinking, “OK, you got me. I used a dumb shortcut.”
But then the professor asked us to look at all the examples on the board and see if we noticed any patterns.
Somebody suggested that the totals seemed to fit a formula based on the number of elements in the set.
The teacher’s response?
“Great observation! Now let’s work together to find the convincing argument.”