June 5, 2010

Proof That's Obvious

My Mom's Dad, an engineer, tells the story of a college Calculus professor who would begin demonstrating a proof on the chalkboard, get half way through, state that he could skip the final steps because "That's obvious!", and then promptly erase the board. I'm sure he was right. I'm sure it was obvious. But it wasn't to his students.

Most High School Geometry teachers largely avoid teaching "proofs", but I've always told students - if we didn't have all these nice beautiful rules about angles, triangles and circles, we'd invent some other way to teach you all how to argue. I love logic, and arguing, but we all get by on faith to some degree or another.

Does physical sensation prove the physical realm is real, or might we all be in the Matrix? Yes, that's silly, but some will argue the point strenuously. I choose to trust my own senses. I believe they're reliable. That's obvious. At least, to most people. But how much else is "obvious"?

At whatever level of Mathematics, it's common for students to "see" a solution without knowing how they did so, or without being able to prove it's correct. Academically, that makes their solution invalid. Practically, it doesn't matter. Someone's proposed solution is every bit as sound as their vision is keen. It just doesn't always communicate.

One of life's frustrations is our struggle to share vision with others when the proof isn't obvious to them. One of life's challenges is to question whether we ourselves are correct about things when others don't won't or can't follow our reasoning.

One of life's ironies is that simpletons can sometimes be correct about things scholars vehemently deny. One of life's mysteries is that any of us may be correct or incorrect about many things, and never know which for certain. And yet somehow God does not feel it necessary to settle all our debates.

Apparently. (But I mean, isn't that obvious?)

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"If I have ever made any valuable discoveries, it has been owing more to patient observation than to any other reason."

-- Isaac Newton