Katara is taking geometry this year, and is having the usual problem of someone studying axiomatic proofs for the first time: the theorems are all obvious, but to understand what is a proof you have to detatch your spatial intuition from the statements.
This is something mathematicians have to learn to do. “Why is this not obvious” is a legitimate mathematical question, one that mathematicians often ask.
Yesterday Katara mentioned the theorem that a line, and a point not on that line, determine a plane and asked why this is something that has to be proved.
Well, I said, you have this postulate that any three non-colinear points determine a plane. So you have a point and a line, and you pick two points on the line, and with the third point, that determines a plane. Now you pick two different points on the line, and with the third point, that also determines a plane. But it's the same plane! It doesn't matter which two points you pick, you always get the same plane. That's interesting, and it's a special property of the particular siutation. There's this rather complicated relationship between all the parts.
Another way to approach this kind of thing, hard to do at this stage but easier later on, is to learn that many things that seem obvious to the senses turn out to be false in more complex situations that our senses are not accustomed to. Katara was excited yesterday to learn about skew lines. This is a great example of something that does not exist in lower dimensions. I said “suppose you were a two-dimensional person, and someone told you that there could be two lines that are not parallel, but they don't intersect. You'd be puzzled, right? But to a three-dimensional person, it's clear.”
“Okay, and you know that any two planes intersect in a straight line, right?”
“In four dimensions, that's false. You can have two planes that intersect in a single point.”