Here is an example of a supposed paradox caused by using Keyne’s Principle of Indifference, which Stove and others (and myself) call the Statistical Syllogism (itself deduced from the symmetry of logical constants). This example is due to Van Fraassen, as quoted in Probability, Causes, and Propensities in Physics, edited by Mauricio Suarez, Chapter 1 (I have a preprint and don’t know the official page numbers).
Incidentally, I thoroughly investigate this in this year’s must-have Christmas present Uncertainty.
Consider a factory that produces cubes of length l up to 2 centimeters. What is the probability that the next cube produced has an edge ≤ 1 cm? A straightforward application of the principle of indifference yields probability = 1/2. But, we could have formulated the question in several different ways. For instance, what is the probability that the next cube has sides with an area ≤ 1 cm^2 ? The principle now yields the answer 1/4. And how about the probability that the next cube has volume ≤ 1 cm^3 ? The answer provided by the principle is now 1/8. These are all inconsistent with each other since they ascribe different probabilities to the occurrence of the very same event.
There is a hidden premise here about infinities that causes the error of supposing there is a contradiction. Let’s see how.
Now real cubes will be of a certain length, a length that will be a discrete and finite number. This applies at least to the way we can measure such cubes, if not to the physical properties of the cube in actuality. Even our finest measurement equipment and procedures can only discern observations to a certain discrete and finite level. This will always be so no matter how inventive the equipment becomes. Measuring to infinite precision would require infinite capacity, which is forever impossible in practice (for us).
Call the smallest unit at which we can measure u. In practice it might be that we cannot measure uniformly, such that on some things we can in places only measure to a x u (a times u), where a > 1, or whatever, but this does not change the analysis, as you shall see.
Any real cube will be c x u in length, where c is an integer greater than or equal to 1. Again, this restriction applies to our knowledge of real cubes as measured by our finest tools. You don’t yet have to believe it applies physically to the cube.
Given what we know so far, and recalling all probability is conditional on what we accept as known, the probability the factory produces cubes with lengths less than our equal to 1 cm (with 2 cm as the maximum), is either 1/2 or close to it. If (c x u)/2 is an integer, then the probability is 1/2; if (c x u)/2 is not an integer the probability will be different. We can’t decide which until we make another assumption.
Does the manufacturing process produce cubes of all sizes? In units of u? Or at certain fixed lengths? Of course there will be leeway: we can probably measure to finer lengths than we can reliably and without fail manufacture cubes; plus not all cubes will be made to the same exact lengths, down to u.
However, it really doesn’t matter. If the machines can’t make cubes down to u, it can make then to some other level of precision, such as 100 x u, which we can then refine as u’ = 100 x u, and then drop the prime mark.
Suppose u = 0.2 cm, then cubes can be 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, or 2 cm in length. The probability a cube with length less than or equal to 1 cm is 1/2. Suppose instead u = 2/13 cm, then cubes can be 2/13, 4/13, etc., up to 26/13 = 2 cm. The probability, given this new information, of lengths ≤ 1 cm is 6/13 < 1/2. The length of 6.5/13 is impossible; impossible to know, anyway, and perhaps impossible in reality.
Now, given these two scenarios or premises, the probability of side surface areas of ≤ 1 cm^2 is again 5/10 = 1/2 and 6/13 (do the math if you don’t see it). The probability of cube areas of ≤ 1 cm^3 is the same. The only possible cubic areas in the latter case are 0.003641329, 0.029130633, 0.098315885, 0.233045061 cm^3 and so on. The values between for example 0.098315885 and 0.233045061 are at least impossible to know, and maybe impossible in actuality.
You can immediately see that Van Fraassen’s objection disappears when u = p/q, and p and q are integers. How did the objection, and subsequent paradox, ever appear in the first place? All paradoxes arise because of the same reason: lurking implicit or unknown premises. The same is true here.
That tacit, and probably unthought of, premise is about infinity. Van Fraassen thought it was possible cubes can be infinitely subdivided in actuality, and not just potentially. If so, then the paradox holds.
But it isn’t true cubes can in actuality be subdivided forever. Of course, the u used above were crude, chosen for simplicity. Real cubes will be finer than 0.2 cm. So let u = p/q = 1/q (p = 1 results in no loss of generality), and let q climb. Let it grow! Let is be a googol (10^100) raised to a googol raised to a googol a googol times, and raise all that to a googol more googols. The result will be a very small u indeed! But one which is still finite and results in discrete units and no paradox. Cubes constructed in this way will still be in blocks of u, and there can be no paradox in the probabilities.
If this u is still “too large” for you, even though it is “practically infinite”, let q grow some more. Do as many raising to the googol googol times the answer as before, and do this every second for a billion years. This will give a q so large you can’t think of it—nobody can think of it!—and a u small but finitely graded. And no paradox.
There are no real restrictions using finite and discrete numbers. The paradox only comes in supposing we carry the process to infinity. It is only at infinity that trouble emerges. At infinity, that strange and imponderable place. Before infinity, there is never a problem, and probability survives without difficulties.
So it must be something strange about infinity. Which is a very true statement.
Now real cubes I claim are also composed of finite and discrete blocks, and cannot in actuality be constructed to infinite precision. We already agree our knowledge is discrete and finite, but I say real cubes are like our knowledge and cannot be infinitely subdivided, except potentially. My proof (in brief) is that everything would then be infinite in this way, which is an explosive idea.
But ignore me, and say I’m wrong. It is still true that no paradox exists unless we are at infinity, a mighty strange land! And then I ask you: what actual evidence have you that things are infinitely decomposable? Your evidence is not (I say) based on any observation anybody has ever made, or could make. You even can only be metaphysical, and then it bumps up against my proof.
No, infinity, though mathematicians toss is around with ease, is too heavy to carry in actuality. Nobody really understands what happens there. And since probability is epistemological, it is no wonder it breaks in just the same way our other thoughts of infinity fracture.
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