Chapter 11

# THE CALCULUS OF PROBABILITIES

February 1, 2022

What has that calculus to do with physical science? The questions I shall raise—without, however, giving them a solution—are naturally raised by the philosopher who is examining the problems of physics. So far is this the case, that in the two preceding chapters I have several times used the words “probability” and “chance.”

“Predicted facts can only be probable.”

are others still, but we cannot detect them; and yet we must make up our minds and adopt a definitive value which will be regarded as the probable value; and for that purpose it is clear that the best thing we can do is to apply Gauss’s law. We have only applied a practical rule referring to subjective probability. And there is no more to be said.

Yet we want to go farther and say that not only the probable value is so much, but that the probable error in the result is so much. This is absolutely invalid : it would be true only if we were sure that all the systematic errors were eliminated, and of that we know absolutely nothing. We have two series of observations; by applying the law of least squares we find that the probable error in the first series is twice as small as in the second. The second series may, however, be more accurate than the first, be- cause the first is perhaps affected by a large systematic error. All that we can say is, that the first series is prob- ably better than the second because its accidental error is smaller, and that we have no reason for affirming that the systematic error is greater for one of the series than for the other, our ignorance on this point being absolute.

VII. Conclusions

In the preceding lines I have set several problems, and have given no solution. I do not re-science and hypothesis gret this, for perhaps they will invite the reader to reflect on these delicate questions. However that may be, there are certain points which seem to be well established. To undertake the calcula- tion of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which has always something arbitrary about it. In the choice of this con- vention we can be guided only by the principle of suffi- cient reason. Unfortunately, this principle is very vague and very elastic, and in the cursory examination we have just made we have seen it assume different forms. The form under which we meet it most often is the belief in continuity, a belief which it would be difficult to justify by apodeictic reasoning, but without which all science would be impossible. Finally, the problems to which the calculus of probabilities may be applied with profit are those in which the result is independent of the hypothe- sis made at the outset, provided only that this hypothesis satisfies the condition of continuity.