> 2 points establishes local linearity, not global linearity. f(x) = x^2 looks linear if you take any 2 arbitrary xs, add a third and you can no longer draw one line that intersects all 3 points.

Yes, certainly. And 3 points can show you a pattern that looks quadratic but isn't, and so on. More to the point, 3 points can rule out a claim of linearity, but they also cannot demonstrate a claim of linearity (e.g., you can usually find 3 collinear points on the graph of a cubic). My point was not to make a mathematical claim myself, but rather to argue that the claim "the minimum number of elements necessary for a pattern is three" is so vague to, I think, be nearly meaningless—and, in particular, unfalsifiable; in my mind, it doesn't deserve the adverb "mathematically".