Coincidence 4: information

E.T. Jaynes was a mathematician and philosopher who, in the twentieth century, did perhaps the most to counter and wrongfoot the frequentist tradition in possibility theory. Jaynes tried to prove that the possibility calculus is rooted in logic – that it is, indeed, as Laplace said, “the calculus of inductive reasoning” – of which random experiments are merely a subset. In other words, Jayne tried to harden the hearts of all who were interested in probability against the idea that probability represented some objective property of objects – or a Popper put it, a propension. To Jayne’s mind, at the same time that the frequentist line attempted to demonstrate that probabilty was something objective, instead of subjective, it also abstracted, absurdly, from the laws of physics. His central case for this was the discourse around coin tossing. Coins, as Jayne points out, are physical objects, and their rise and fall is completely described by the physics of ballistics. (I take this example from Jayne’s book, Probability theory: the logic of the sciences). Thus, to say that a coin with heads and tails has a fairly equal chance of landing on either side, with a lean a bit to heads over a long series of tosses, is to speak nonsense. Rather, everything depends on how a coin is tossed, as a physical object.

The laws of mechanics now tell us the following. The ellipsoid of inertia of a thin disc is

an oblate spheroid of eccentricity 1/√2. The displacement x does not affect the symmetry of this ellipsoid, and, so according to the Poinsot construction, as found in textbooks on rigid dynamics (such as Routh, 1905, or Goldstein, 1980, Chap. 5), the polhodes remain circles concentric with the axis of the coin. In consequence, the character of the tumbling motion of the coin while in flight is exactly the same for a biased as an unbiased coin, except

that for the biased one it is the center of gravity, rather than the geometrical center, which describes the parabolic ‘free particle’ trajectory.”

Given these physical facts, this is what Jayne suggests:

Therefore, in order to know which face will be uppermost in your hand, you have only

to carry out the following procedure. Denote by k a unit vector passing through the coin

along its axis, with its point on the ‘heads’ side. Now toss the coin with a twist so that k and

n make an acute angle, then catch it with your palm held flat, in a plane normal to n. On

successive tosses, you can let the direction of n, the magnitude of the angular momentum,

and the angle between n and k, vary widely; the tumbling motion will then appear entirely

different to the eye on different tosses, and it would require almost superhuman powers of

observation to discover your strategy.

Thus, anyone familiar with the law of conservation of angular momentum can, after some

practice, cheat at the usual coin-toss game and call his shots with 100% accuracy.”

Jayne’s point is that probability is not a spooky physical property connected with the two sidedness of the coin, but is a logical abstraction describing the physical event, including in its reference set the manner of the tossing.

Jayne goes on to demolish other examples from the frequentist literature. Here’s his conclusion:

“… those who assert the existence of physical probabilities do so in the belief that this establishes for their position an ‘objectivity’ that those who speak only of a ‘state of knowledge’ lack. Yet to assert as fact something which cannot be either proved or disproved by observation of facts is the opposite of objectivity; it is to assert something that one could not possibly know to be true. Such an assertion is not even entitled to be called a description of a ‘state of knowledge’.”

This conclusion led Jaynes to some radical and unorthodox positions. In particular, it led him to stress lack of knowledge, rather than physicalism, when accounting for quantum mechanics. He is famous for applying this, as well, to thermodyamics:  “entropy is an anthropomorphic concept, not only in the well known statistical sense that it measures the extent of human ignorance as to the microstate. Even at the purely phenomenological level entropy is an anthropomorphic concept. For it is a property not of the physical system but of the particular experiments you or I choose to perform on it.”

Often, while following a philosophical train of thought, one encounters a moment when the values one has been using strangely seem to inverse themselves. It is like the child's game of closing your eyes and spinning around and around: at the moment you stop and open your eyes, it seems that it is the world that is spinning around and around and you are standing still in the eye of it. The argument about probability partakes of that vertigo. The classical school inherits from Laplace the confidence that the world is a totally determined system, in which all phenomena can eventually traced back to material causes. And yet, to get to this argument, the school has to advance the thesis that probability is simply a measure of knowledge - or, to use the modern term, information. This means that, in classical terms, possibility is subjective. On the other side is the world picture that rejects crude determinism and accords chance a very real place. This school, then, takes possibility as as a real property, or in Popper's terminology, propensity, of events. This is, ultimately, an argument that makes possible an ontologically distinct thing called subjectivity. But, in grounding subjectivity in chance, in making possibility objective, this school entangles itself in all the logical problems adduced by Jaynes. And so, as the first group bases its determinism, which ultimately dissolves subjectivity, on the subjectivity of the probability calculus, the other group bases its indeterminism on the reification of a spooky non-cause. As I've pointed out, what goes for chance goes for coincidence. Perhaps here a Kantian probabilist could claim that we have reached the limit of our reason - the antinomies of chance are undecidable. But I'm pretty sure Jaynes would question whether, ultimately, we are not just making undecidable a case of our lack of knowledge, thus forcing us back towards his school.