Dangers of extrapolation

The naïve conception of the figure of the Earth as a plane is locally correct everywhere on Earth but globally false. Observers in every place on Earth see evidence of a finite plane that ends at the horizon. The simplest generalization of these observations is to say they are all seeing the same plane of the Earth. So the naïve conception is empirical. But extrapolating these observations leads to a false generalization.

What is wrong? It is fine as a starting point but it should be challenged. Consider new kinds of evidence, not just more of the same kind of evidence. Also, consider other conceptual schemes. The concept of a spherical Earth was first proposed on aesthetic grounds by ancient Greeks who held the sphere as the simplest, most perfect form. Given these two rival conceptions, a way is needed to judge between them. The empirical way is to look for evidence for one that is evidence against the other.

The naïve conception illustrates the dangers of extrapolation. It is so easy to extrapolate but it can be so wrong. Extrapolation often works to a limited degree but increasingly diverges until it is simply mistaken. Extrapolation is like an analogy: it is valid to a point but rarely valid at all points. We should always look for the limits of extrapolation.

What is a guideline for the limits of extrapolation? Consider a time series such as mean temperature for 100 years. Could we extrapolate it 1000 years? That would be much longer than the original time series which seems excessive. How about 100 years? Maybe, though that would be pushing it. For 10 years? That seems reasonable. So extrapolation for a period less than the original data is about the most that can be expected.

Every observation has a date in which it took place. The range of all observation dates is the range of recorded history, which is less than 10 thousand years. This is a problem for natural history which extrapolates to time periods much longer than the period of observation. The solution is to assign risk to extrapolations: the greater the extrapolation, the greater the risk of error. Unfortunately, this is not done; in natural history the risk of error from extensive extrapolation is simply ignored.

May 2008