Lovegrove Mathematicals

"Dedicated to making Likelinesses the entity of prime interest"

Originally put forward by van Fraasan (1989), there are several versions, all essentially the same, of the Perfect Cube Factory. The purpose was to demonstrate that the principle of indifference could give rise to contradictions and so was not a valid form of argument.

This version, which I give because I consider it to be one of the clearer versions, is taken from Hájek (2012).

A factory produces cubes with side-length between 0 and 1 foot; what is the
probability that a randomly chosen cube has side-length between 0 and 1/2 a
foot? The tempting answer is 1/2, as we imagine a process of production that
is uniformly distributed over side-length. But the question could have been
given an equivalent restatement: A factory produces cubes with face-area
between 0 and 1 square-feet; what is the probability that a randomly chosen
cube has face-area between 0 and 1/4 square-feet? Now the tempting answer is
1/4, as we imagine a process of production that is uniformly distributed
over face-area. This is already disastrous, as we cannot allow the same
event to have two different probabilities (especially if this interpretation
is to be admissible!). But there is worse to come, for the problem could
have been restated equivalently again: A factory produces cubes with volume
between 0 and 1 cubic feet; what is the probability that a randomly chosen
cube has volume between 0 and 1/8 cubic-feet? Now the tempting answer is
1/8, as we imagine a process of production that is uniformly distributed
over volume. And so on for all of the infinitely many equivalent
reformulations of the problem (in terms of the fourth, fifth, … power of the
length, and indeed in terms of every non-zero real-valued exponent of the
length). What, then, is *the* probability
of the event in question?

The paradox arises because the principle of indifference can be used in incompatible ways. We have no evidence that favors the side-length lying in the interval [0, 1/2] over its lying in [1/2, 1], or vice versa, so the principle requires us to give probability 1/2 to each. Unfortunately, we also have no evidence that favors the face-area lying in any of the four intervals [0, 1/4], [1/4, 1/2], [1/2, 3/4], and [3/4, 1] over any of the others, so we must give probability 1/4 to each. The event ‘the side-length lies in [0, 1/2]’, receives a different probability when merely redescribed. And so it goes, for all the other reformulations of the problem. We cannot meet any pair of these constraints simultaneously, let alone all of them.

The argument is based on there being a contradiction. The precise description of that contradiction varies according to the way the author has described the problem, but the reason for its existence is always the same.

We seem to have two ways to calculate the distribution of face areas. These are:-

1. To apply the Principle of Indifference directly to the distribution of areas, to obtain
the uniform distribution.

2. To apply the Principle of Indifference to the distribution of side-lengths, so that *
they* follow the uniform distribution, and then
(because area=side^{2})
extrapolate to the faces by squaring it. This
gives the face-areas a non-uniform distribution.

(To do these, we need to subdivide the intervals of lengths/areas/volumes into smaller subintervals. That a cube might have its length, etc, falling into a subinterval is one of the "possibilities" referred to by Insufficiency. The diagram shows 12 such subintervals.)

So, the argument goes, there is a contradiction between the extrapolation and Indifference,
and one (or both) of them must therefore be wrong. Since it cannot
be (?) that the method of extrapolation is wrong -face areas
*are* the square of
sides- the error must be with Indifference.

As shown in the diagram, a similar contradiction is obtained when considering volumes.

A good place to start unravelling what is going on is with Hájek's question "What,
then, is *the* probability of the event in question?". The answer to this is "We don't know: we would
have to visit the factory and measure some cubes to know that."

Once we have answered that question, there follows another: "What, then, is the uniform distribution giving us?". The answer to this is that it is giving us an estimate -actually, the so-called best-estimate- of the distribution. This is the fundamental error in the problem: it confuses the probability with the best-estimate of the probability.

The uniform distribution of side-lengths is an *estimate*: an *approximation* which comes about because of a theoretical averaging process, usually making
use of a symmetry argument. It is an average over the set of all possible distributions. Likewise for the uniform distributions of
face-areas or of
cube-volumes.

When we extrapolate from the uniformly-distributed estimated side-lengths, by squaring or cubing as appropriate, do we perhaps
then obtain the actual distribution? We do not. We obtain
another estimate.

So we have two different estimates
of face-area, obtained in two different ways: one, by best-estimating them
directly, the other by best-estimating the sides and then squaring that.
Likewise for volumes (but involving cubing rather than squaring).

This is what is claimed to be the contradiction. However, there is
nothing wrong with having two different estimates: there is
no logical contradiction of the form A & not-A, and having different
estimates of the same thing is part of everyday science. The paradox arises
only when we stop using the two different estimates as estimates and start
treating them both as if they were the actual distribution: it is here that the contradiction arises, *
the distribution is uniform & the distribution is not uniform*.

In summary, The Perfect Cube Factory goes wrong by claiming that Indifference gives the actual probability
rather than an estimate of, or mean of, the probability. The
Indifference *Theorem*,
however, does not give probabilities but likelinesses, that is the mean,
over the set of all distributions. It says that the mean (over the set of
all possible distributions) is uniform: this applies to the mean of the
distribution of side-lengths, also to the mean of the distribution of
face-areas and to the mean of the distribution of volumes.

There is an alternative approach which uses the Combination Theorem rather than Indifference.

Divide the interval [0,1] into N subintervals (in the diagram, N=12). Use those subtintervals to define some distribution of degree N. That distribution may be any element of S(N).

If we knew the distribution then its values would be the probabilities we seek, but we do not know it. However, we do know that the underlying set is S(N) and that we have no data, so the given histogram is 0. So the Combination Theorem gives each subinterval a likeliness of 1/N, ie the likelinesses follow the uniform distribution.

But we have not said whether we are talking about the distribution of sides, or of face-areas or of volume, so the same argument applies to all three.

So all three distributions of likelinesses follow the uniform distribution.