Probability in the Physical Sciences

I define the “second degree of ignorance” as us knowing the law, but not knowing the initial state of the system.

An example is: What is the probable present distribution of the minor planets on the zodiac?

We know they obey the laws of Kepler.

We may even, without changing the nature of the problem, suppose that their orbits are circular and situated in the same plane, a plane which we are given. On the other hand, we know absolutely nothing about their initial distribution.

However, we do not hesitate to affirm that this distribution is now nearly uniform. Why?

Let b be the longitude of a minor planet in the initial epoch—that is to say, the epoch zero.

Let a be its mean motion. Its longitude at the present time—i.e., at the time t will be at + b.

To say that the present distribution is uniform is to say that the mean value of the sines and cosines of multiples of at + b is zero.

Why do we assert this?

Let us represent our minor planet by a point in a plane—namely, the point whose co-ordinates are a and b. All these representative points will be contained in a certain region of the plane, but as they are very numerous this region will appear dotted with points. We know nothing else about the distribution of the points.

Now what do we do when we apply the calculus of probabilities to such a question as this?

What is the probability that one or more representative points may be found in a certain portion of the plane?

In our ignorance we are compelled to make an arbitrary hypothesis.

To explain the nature of this hypothesis I may be allowed to use, instead of a mathematical formula, a crude but concrete image.

Let us suppose that over the surface of our plane has been spread imaginary matter, the density of which is variable, but varies continuously.

We shall then agree to say that the probable number of representative points to be found on a certain portion of the plane is proportional to the quantity of this imaginary matter which is found there.

If there are, then, two regions of the plane of the same extent, the probabilities that a representative point of one of our minor planets is in one or other of these regions will be as the mean densities of the imaginary matter in one or other of the regions.

Here then are two distributions, one real, in which the representative points are very numerous, very close together, but discrete like the molecules of matter in the atomic hypothesis; the other remote from reality, in which our representative points are replaced by imaginary continuous matter.

We know that the latter cannot be real, but we are forced to adopt it through our ignorance. If, again, we had some idea of the real distribution of the representative points, we could arrange it so that in a region of some extent the density of this imaginary continuous matter may be nearly proportional to the number of representative points, or, if it is preferred, to the number of atoms which are contained in that region.

Even that is impossible, and our ignorance is so great that we are forced to choose arbitrarily the function which defines the density of our imaginary matter.

We shall be compelled to adopt a hypothesis from which we can hardly get away; we shall suppose that this function is continuous.

That is sufficient, as we shall see, tothe calculus of probabilities.

What is at the instant t the probable distribution of the minor planets—or rather, what is the mean value of the sine of the longitude at the moment t—i.e., of sin(at + b)? We made at the outset an arbitrary convention, but if we adopt it, this probable value is entirely defined. Let us decompose the plane into elements of surface.

Consider the value of sin(at + b) at the centre of each of these elements. Multiply this value by the surface of the element and by the corresponding density of the imaginary matter. Let us then take the sum for all the elements of the plane.

This sum, by definition, will be the probable mean value we seek, which will thus be expressed by a double integral. It may be thought at first that this mean value depends on the choice of the function φ which defines the density of the imaginary matter, and as this function φ is arbitrary, we can, according to the arbitrary choice which we make, obtain a certain mean value. But this is not the case.

A simple calculation shows us that our double integral decreases very rapidly as t increases. Thus, I cannot tell what hypothesis to make as to the probability of this or that initial distribution, but when once the hypothesis is made the result will be the same, and this gets me out of my difficulty.

Whatever the function φ may be, the mean value tends towards zero as t increases, and as the minor planets have certainly accomplished a very large number of revolutions, I may assert that this mean value is very small. I may give to φ any value I choose, with one restriction: this function must be continuous; and, in fact, from the point of view of subjective probability, the choice of a discontinuous function would have been unreasonable.

Why should I suppose that the initial longitude might be exactly 0 ◦ , but that it could not lie between 0 ◦ and 1 ◦ ?

The difficulty reappears if we look at it from the point of view of objective probability; if we pass from our imaginary distribution in which the supposititious matter was assumed to be continuous, to the real distribution in which our representative points are formed as discrete atoms. The mean value of sin(at + b) will be represented quite simply by

1 X sin(at + b), n

n being the number of minor planets. Instead of a double integral referring to a continuous function, we shall have a sum of discrete terms. However, no one will seriously doubt that this mean value is practically very small. Our representative points being very close together, our dis- crete sum will in general differ very little from an integral.

An integral is the limit towards which a sum of terms tends when the number of these terms is indefinitely increased. If the terms are very numerous, the sum will differ very little from its limit—that is to say, from the integral, and what I said of the latter will still be true of the sum itself. But there are exceptions. If, for instance,

π − at, the longitude of for all the minor planets b = 2 π all the planets at the time t would be , and the mean 2

value in question would be evidently unity.

For this to be the case at the time 0, the minor planets must have all been lying on a kind of spiral of peculiar form, with its spires very close together. All will admit that such an initial distribution is extremely improbable (and even if it were realised, the distribution would not be uniform at the present time—for example, on the 1st January 1900;

but it would become so a few years later). Why, then, do we think this initial distribution improbable?

This must be explained, for if we are wrong in rejecting as improbable this absurd hypothesis, our inquiry breaks down, and we can no longer affirm anything on the subject of the probability of this or that present distribution. Once more we shall invoke the principle of sufficient reason, to which we must always recur. We might admit that at the beginning the planets were distributed almost in a straight line.

We might admit that they were irregularly distributed. But it seems to us that there is no sufficient reason for the unknown cause that gave them birth to have acted along a curve so regular and yet so complicated, which would appear to have been expressly chosen so that the distribution at the present day would not be uniform.