The Paradox of Induction
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Extrapolation and interpolation
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One particular form of the paradox of induction concerns the process in science of analysing results by fitting curves to data points. This results in the extrapolation and interpolation of curves and lines. Extrapolation means to extend a line or pattern beyond the range of the data provided by the experiment. It is illustrated by the following graph. ... We have a relationship between two variables X and Y and we have plotted a graph of the data, and drawn a straight line. The question is can we now continue the relationship outside the range of values found, and if we do so, should be continue it by means of a straight line? To extrapolate means to carry on a linear relationship outside the given range of values. For example, ... The dotted line represents the extrapolation. However, we do not know for sure that the relationship can be continued in this way; there are any number of alternative possibilities. ... This situation occurs in practical cases all the time. For example, if X was the load on a wire and Y was its extension, then after a point the extension would cease to be proportional to the load and we would see precisely this kind of graph. Interpolation is concerned with the problem of fitting a line or relationship between data points. Suppose we start with the following data points. Strictly speaking we have not made observations in the ranges between the data points. We might naturally assume that this was a straight-line relationship, and to fit a straight line to the data is to interpolate between them. However, there are many other relationships that could be fitted to the same data points.
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Contents of
The Paradox of Induction
1 Prescriptive philosophy of science
2 The problem or paradox of induction
3 Hume and the formulation of the paradox of induction
4 Attempts to solve the paradox of induction
5 The paradox of induction and the claim that probability is all we ought to seek
6 Swinburn and confirmation theory
7 Falsificationism and the paradox of induction
8 Extrapolation and interpolation
9 Occam razor and the paradox of induction
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