# Following an idea to its logical conclusion

Following an idea to its logical conclusion might be extrapolating a model beyond its valid range.

Suppose you have a football field with area A. If you make two parallel sides twice as long, then the area will be 2A. If you double the length of the sides again, the area will be 4A. Following this reason to its logical conclusion, you could double the length of the sides as many times as you wish, say 15 times, and each time the area doubles.

Except that’s not true. By the time you’ve doubled the length of the sides 15 times, you have a shape so big that it is far from being a rectangle. The fact that Earth is round matters a lot for figure that big.

Euclidean geometry models our world really well for rectangles the size of a football field, or even rectangles the size of Kansas. But eventually it breaks down. If the top extends to the north pole, your rectangle becomes a spherical triangle.

The problem in this example isn’t logic; it’s geometry. If you double the length of the sides of a Euclidean rectangle 15 times, you do double the area 15 times. A football field is not exactly a Euclidean rectangle, though it’s close enough for all practical purposes. Even Kansas is a Euclidean rectangle for most practical purposes. But a figure on the surface of the earth with sides thousands of miles long is definitely not Euclidean.

Models are based on experience with data within some range. The surprising thing about Newtonian physics is not that it breaks down at a subatomic scale and at a cosmic scale. The surprising thing is that it is usually adequate for everything in between.

Most models do not scale up or down over anywhere near as many orders of magnitude as Euclidean geometry or Newtonian physics. If a dose-response curve, for example, is linear for based on observations in the range of 10 to 100 milligrams, nobody in his right mind would expect the curve to remain linear for doses up to a kilogram. It wouldn’t be surprising to find out that linearity breaks down before you get to 200 milligrams.