Wednesday, February 14, 2007

Curve-Fitting and Description-Dependence

I started thinking about this after Aidan Lyon's excellent talk on the curve-fitting problem yesterday.

Graham Priest in his 1976 article Gruesome Simplicity (this link is to JSTOR) discusses curve-fitting as a way of making inductive inferences. When we plot observed values of two related quantities x and y on a graph, we have several options for which curve to draw between them. The simplicity of the curve has to be traded off against fit with the existing data points, and it is a taxing problem to say how best this should be done. Yet we often do think we can choose an appropriate curve, and use it to make predictions concerning as-yet-unobserved values of x and y.

What Priest shows is that 'certain very natural transformations' on data sets result in different curves appearing to be 'best' and correspondingly conflicting predictions being delivered. Priest therefore claims to have shown that 'which prediction is best depends not on the situation but how you describe it. (Equivalent descriptions do not give the same answers.)' (p. 432). This sort of description-dependence sounds unsettling; we would like our predictions to be sensitive only to our data, and not affected by accidental features of the ways we happen to represent that data.

It seems to have been accepted in the subsequent literature that Priest's problem, if it cannot be avoided, establishes a worrying kind of description-dependence. But in my opinion the existence of such description-dependence is not established by Priest's argument. To get that conclusion, we would need an additional premise: that when we perform the transformations on the data that generate the new predictions, we are just redescribing the same situation, as opposed to considering a different situation.

What counts as a different situation? Well, for these purposes, we should consider any difference which is not merely a difference in our descriptions a difference in the situation described. (If this were not what Priest had in mind, i.e. if there were room for something else to differ besides the situation and our descriptions of it, it would not follow that a difference in prediction without a difference in the situation described must be due merely to a difference merely in our descriptions.)

There is reason to suppose that many 'transformations' of data – even very simple a priori ones – lead to description of a different situation. A priori inference, for example, enables me to conclude from This plane figure has three angles that This plane figure has three sides, but these are two different propositions involving different properties. The difference between these two propositions is (plausibly) not merely redescription; what's being described – which language-independent properties are being talked about – differs in each case.

Similarly, turning to one of Priest's own examples, when we transform data about velocity and momentum into data about velocity and kinetic energy, there is a difference in what is described and not merely a difference in description. Momentum and kinetic energy are related by the equation E = pv/2, but they are not therefore the same thing.

This might help make Priest's result a little less unsettling than Priest himself suggests. It is less philosophically disturbing to conclude that thinking about two different situations can lead us to make two different predictions than to conclude that describing the same situation two different ways can lead us to make two different predictions.

That's not to say it makes the result entirely comfortable; the difference in predictions is still somewhat disturbing, given that we know the transformation relationship between the two situations. It may well be interesting and difficult, to say what (if anything) we ought to predict in these circumstances. But it has not yet been established that the problem arises due to description-dependence.