Sometimes Jesus us calls us to do things to accomplish his plan for which we must resist our human experience. This lesson can be seen as a principle in John 15:1-11. Jesus calls us Peter to cast the nets into waters that have been fished, unsuccessfully. Peter knew it wouldn’t work and yet, Jesus used his obedience to teach him to trust.
In mathematics, there are countless examples of steps that seem counter intuitive and yet the end result is that by going in the opposite direction, we can greatly simplify the problem. The simplest example that comes to mind is in algebra where you are simplifying either rational expressions or fractional expressions that have roots in the numerator or denominator. To simplify, you start making things more complicated by multiplying in terms in the numerator or denominator. The end result is a simpler expression.
I have always felt that proofs by
contradiction contrapositive fit in this category as well. When one first starts learning proof techniques this especially seems counter-intuitive. Of course, most of us math guys feel the direct proof, when available, is a “better” proof, even if it’s clumsier. But it’s still logically equivalent.
Updated (10/02/06): I readily admit my mistakes, especially since I make them so rarely :). The last paragraph above should have referred to proof by contrapositive, not contradiction. Although, proofs by contradiction also seem counter-intuitive. At least, I know someone is reading.
where is a contradiction.