Alternatives and Models

A further problem that arises in the use of any test that simply rejects a hypothesis without at the same time considering possible alternatives[...] Is it of the slightest use to reject a hypothesis until we have some idea of what to put in its place? If there is no clearly stated alternative, and the null hypothesis is rejected, we are simply left without any rule at all, whereas the null hypothesis, though not satisfactory, may at any rate show some sort of correspondence with the facts. It may, for instance, represent 90% of the variation and to that extent may be of considerable use in prediction, even though the remaining 10% may be larger than we should expect if it was strictly true. 

Consider, for instance, the history of the law of gravitation. Newton first derived it from Kepler’s laws and a comparison of the accelerations of the moon and of a body falling freely at the earth’s surface. Extending it to take account of the mutual attractions of the planets and of the perturbations of the moon by the sun, he got the periods and orders of magnitude of the principal perturbations. But he did not explain the long inequality of Jupiter and Saturn, with a period of 880 years, which gives displacements in longitude of 1196″ and 2908″ of arc for the two planets, and was only explained by Laplace a century later. 

The theory of the moon has been taken only in the present century, by E. W. Brown, to a stage where most of the outstanding errors of calculation can be said to be within the errors of observation; there are discrepancies between observation and calculation that are attributed to changes of the rotation of the earth; but these discrepancies are our principal ground for believing in the existence of these changes. In fact agreement with Newton’s law was not given by the data used to establish it, because these data included the main inequalities of the moon; it was not given during his lifetime, because the data included the long inequality of Jupiter and Saturn; and when Einstein’s modification was adopted the agreement of observation with Newton’s law was 300 times as good as Newton ever knew. 

Even the latter appears at present as powerless as Newton’s to explain the long empirical term in the moon’s longitude and the secular motion of the node of Venus. There has not been a single date in the history of the law of gravitation when a modern significance test would not have rejected all laws and left us with no law. -- Theory of Probability, Harold Jeffreys 1939.