The Principle of Indifference: II
random chords
I sometimes wonder what 19th century mathematicians were smoking. The problems they come up with are often... a little weird. But they can serve as a warm-up to somewhat less weird questions about why perpetual motion machines don't work.
French mathematician Joseph Bertrand came up with the following problem:
The set-up is illustrated in the image at the top of this post. The black circle has an inscribed equilateral triangle in grey. The red lines are chords: a "chord" in this context is just a line that spans the circle. The blue lines are not chords. They're just lines. We'll come back to them in a bit.
The "generally presented" statement reminds us that this presentation is part of a formal ritual designed to mislead us, and we should first restate the problem in less-misleading language that clarifies what the situation being considered actually is. Like any good magic trick, it's not always obvious where the misdirection is happening, and it often happens well upstream of where we are on the look out for it.
The first part of the ritual statement is no (apparent) problem:
1) Consider an equilateral triangle inscribed into a circle.
That's just constructive geometry. But given the way it's used later on, it's kind of a weird way to specify a length, isn't it? Why not just take some arbitrary length a and ask about it? An inscribed equilateral triangle in a circle of radius R has side a = R*sqrt(3), so why not ask about that? Or R/2, or some other number? Why introduce the entirely unnecessary inscribed equilateral triangle when you can just give a number?
This is where the first piece of misdirection comes in: by focusing on an inscribed triangle, our limited attention is drawn to the inside of the circle. Always be suspicious of information in the statement of a paradox that is specified in a needlessly complex way: such information exists solely as an attention sink, and when dealing with ill-stated questions you need all the attention you can get. So look for ways to restate complex conditions in terms of simpler but no less general ones.
Focusing on the inside of the circle becomes an issue if that's where we start looking for the fundamental object in the next part of the problem, which is:
2) Suppose a chord of the circle is chosen... how?
What on Earth does a "random chord" mean?
A chord is a line, or at least a line segment, which for our purposes is the same thing. A "line" in the language of pure geometry extends off to infinity in both directions. A "line segment" has finite length.
Bertrand suggested three possible methods for constructing a "random chord", all of which are intended to both construct and select lines that are chords. This is important: he mixes the construction of lines with the selection of chords, which are unrelated to each other. Bertrand's proposed methods are:
1) Endpoints: a line segment chosen by taking two random angles and using them to define points on the circle described in the problem, as opposed to any of the other equally good circles one might use.
2) Radial Points: choose a random radius of the circle and a random point along it within the circle, and draw the chord perpendicular to the radius, as opposed to any of the other lines passing through that point at different angles.
3) Midpoints: choose a point within the circle and construct a chord with this point as its midpoint, as opposed to any other point along the chord--say the 1/3 point--you might choose.
Each of these results in a different answer to the question in the ritual statement of the problem, which is sometimes used to argue that the principle of indifference is insufficient to uniquely answer problems like this. But as we saw in the water/wine problem, selecting a uniform distribution over some convenient parameterization of the problem is no guarantee that the physically important quantities will be treated indifferently.
In this case, there is a fundamental ambiguity in the problem statement, and therefore a degree of freedom in how we resolve it. This is not a problem with the principle of indifference, but a problem with the problem.
One could easily argue that "a random chord" should be drawn from a uniform distribution in length, in which case the problem answers itself: the chords range from 0 to 2*R in length, and the probability one of them will have a length greater than a (whatever we happen to choose a to be) will just be (2R-a)/2R, the fraction of the length distribution longer than a.
What we mean by "randomness" has to be carefully specified if we don't want to get ourselves into trouble, and in each of Bertrand's methods we are choosing precisely two parameters from uniform distributions and then using an additional--decidedly non-random--rule to specify one particular line.
There is no warrant for the additional rule in any of these cases. Why choose the rule that Bertrand does when we could choose any number of others? For example, we could use any circle whatsoever on the plane to allow us to use two random angles to generate a random line, some of which would be chords. If the circle we choose happens to be smaller than and interior to the problem circle, all of those lines will be chords. Why use the circle the problem is talking about? The fact that it's mentioned in the problem statement doesn't give us any warrant to use it for this purpose.
This is an aspect of how autistic people view the world: we don't see it as obvious that we should use the circle described in the problem to generate chords when we could equally well use any other circle for the purpose. Neurotypical people fill in certain kinds of blank in a way that is so natural to them that it's difficult for them to see any alternatives, and I've come to believe that this is one of the things that causes paradoxes to persist even when the error in them is obvious to an autistic person.
A "random chord", if it has any meaning at all, has to mean a "random line that happens to be a chord". Bertrand's fundamental error is that he tries to restrict randomness to chords, when "restricted randomness" is a contradiction. A chord is just a segment of a line, and to respect the principle of indifference we have to sample all the possible lines on the plane and then post-select for the ones that are chords. Think of this as dropping the circle anywhere on the plane: the sample of lines that pass through it are chords.
That is, through any point in the plane an infinite number of lines pass (at different angles) and while the number of lines is infinite, the density of lines is constant: for any point in the plane, the same (infinite) number of lines pass through it. It's one of the peculiarities of mathematics that we can talk about the relative sizes of infinities, but we can. And if we use a sampling technique that prefers some of these lines over the others, we're not being indifferent. Sampling just points within a circle of radius 1, this is what ten random lines through each of them looks like:
All angles are equally probable.
Bertrand's methods, however, produce either a single line through each point, or a complicated distribution of lines (in the case of the two-angles method). Both the radius and the midpoint method select one particular line through each point (they differ in how the points are chosen). This selection is done arbitrarily and without warrant: there is no reason whatsoever to declare these particular lines are "special", and doing so violates the principle of indifference, which tells us we must treat all lines the same:
So it turns out that by focusing the discussion on chords rather than lines the ritual statement of the problem leads us down the garden path. Choosing a random chord seems like a complex and difficult process, and focusing on that exhausts our limited attention and allows the paradoxians to fool us.
Only by choosing all line parameters independently and indifferently can we make a fair claim to having chosen a "random chord". If we do anything else, we can identify the rule used to construct the chords by looking at their distribution of parameters, and as we'll see in the weeks to come, "Randomness means never knowing where a value came from" (or something like that).
That is: a random distribution does not carry information about its origins. This is a crucial connection between thermodynamics and information theory, and the fundamental reason why perpetual motion machines don't work.
So how do we choose random lines? There are an infinite number of ways of parameterizing a line, but the density of lines with respect to those parameters is generally non-uniform, which makes them a pain to sample.
Since what I want is a density of lines that is the same at every point, I'm going to sample (x, y) points randomly within the circle, and then sample a random angle to define a line passing through that point. Even though we're sampling only within the circle, this turns out to be equivalent to sampling a uniform density across the whole plane, but the explanation of that involves several hundred words of more-or-less esoteric math I'm assuming most of my readers can do without.
So here is my restatement of the problem:
Consider a circle of radius R. If we construct a sample of chords of this circle that respects the uniform density of lines on the plane, what fraction of those chords will be longer than some length a?
I don't see any way of taking the ambiguity out of the problem without clarifying what a "random chord" means: otherwise the problem statement is too ambiguous to be interesting.
The problem then has a straightforward solution, at least computationally, and we find that for a = R*sqrt(3), which is the side length of Bertrand's inscribed triangle, the probability of getting a chord at least as long is 0.609. As a matter of interest, I tested that this is in fact independent of whether I sample a large area outside the circle--which it is--because the argument that leads to sampling inside it really is quite subtle, and I've been known to make mistakes.
The distribution of chord lengths looks like this:
A certain type of mathematical purist will complain that by merely computing the result I've missed the point, which is to get an analytical (algebraic) expression for the distribution. There are plenty of cases where that's useful and interesting, but life is short, and I don't think it's sufficiently useful or interesting to go into here.
The purpose of this whole exercise is to get clear in our heads that randomness isn't something we get from casually throwing stuff together. It has to be quite carefully constructed, and it's a curious thing that this is what the universe actually does, sometimes in quite surprising ways, as we'll start to see more clearly next week when I move from these contrived mathematical problems to questions of perpetual motion machines and why they don't work.