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Wednesday, August 12, 2015

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.

Monday, August 10, 2015

coincidence 3: the naive and the sophisticated novelist

In 1850, Dickens began a novel with an exemplary sentence: “Whether I shall turn out to be the hero of my own life, or whether that station shall be held by someone else, these pages will tell.”  It was, in fact, obviously the nature of these pages – the novel – to tell this story. It went without saying that for Dickens, as well as for other Victorian novelists, the interest of the novel was tied to interest in the individual. If there was an anxiety here, it was about heroism in Carlyle’s key, a heroism that passes the moral tests of life – but there was no doubt that a life was definitely not a matter determined within a larger social pattern, and only of interest insofar as it could be grouped with a subpopulation in order to display certain tendencies. In this sense, the novel bet everything on the ideology of heroism.

Even so, at the same time, in mid nineteenth century, there were indications that a radically different point of view, the statistical mindset, was winning minds outside the circle of literature.  Quetelet, for instance, in 1835 had already tried to show that crime should not be understood through its individual instances, but through statistics demonstrating its likelihood of incidence. From this, Quetelet inferred that it was society, and not the criminal, which produced crime, just as an orange tree produced oranges. We would not hold an orange responsible being an orange, although we might pluck it and squeeze it to death for its juice – just as we might take down a criminal and cut off his head to satisfy the principles of social hygiene.

Dicken’s notion of the novel and the individual produced what Robert Musil called a naïve, or old fashioned story form, which was very difficult to break with. In his view –a view, it must be admitted, conditioned by Musil’s envy of the fame of the great modernists – Ulysses and A la recherche are still footed in the archaic world of certainty and heroism, instead of the world in which that ground had disappeared and criminals could be considered the fruit of society, rather than bad actors making bad decisions, while characters could be considered as hybrids of the interior thoughts that, they delusively believe, guide them, and the administrative purposes for which they employed by exterior forces.

It is in this context that Musil thought a lot about coincidence – Zufall. Chance, after all, is felt as coincidence in a story, especially when science shatters our confidence that a life and a life story are one and the same thing.  In his diaries,  Musil piled up references to popular work on probability and chance in the field of math and physics. One of his sources was Erwin Schroedinger’s essay on the Gesetz der Zufall – the Law of Chance – in Koralle, a popular science magazine, which appeared in 1928.

It is a small, lucid essay, with two themes. One is that our understanding of the physical world is based not on certainty, but on probability. The other theme is that the second law of thermodynamics, which posits that systems advance from order to larger degrees of disorder, doesn’t free us from the link of determinism, if by determinism we mean unpredictability. Rather, entropy is highly predictable.

To make this point, Schroedinger uses an example that would have struck a writer like Musil – the example of the library.

He asks us to imagine a library that has been organized so that all the books in it are numbered and put in their proper places. And then he imagines a horde coming in on Monday – surely, students right before exam time – and going through the library and taking out books and putting them back with no regard for their proper place:
 Now the astonishing feature is that this process proves to be subject to very definite laws, especially if we suppose that the valumes are taken from the shelves in the same haphazard way as they are put back…. If we suppose that there were eighty volumes of Goethe’s works, for instance, neatly arranged in one section of the library when the casual mob entered, and if we find that only sixty volumes are now in their places while the other twenty are scattered about here and there, then we can expect during the second week about fifteen volumes will disappear from the row, and about eleven volumes will vanish during the third week, etc. For since we have supposed that the books are taken out quite at random, the probability that one of the remaining volumes will meet with this misfortune decreases as their number decreases.”

Schroedinger concedes that his example is stylized – really, for the predictions to be more exact, the numbers must be bigger. If the collected works amounted to 80,000 among millions of volumes, the deviations from the predicted number of remaining books would be smaller.

Schroedinger’s library example is interesting to follow through. If this were a real library, then some of the Goethe volumes would be checked out, and some of the books that were scattered around would be discovered by library assistants and put back in their place. In terms of the second law, what this would mean is that the system had feedbacks – which means that it is not entirely closed.
“ We do not wish to asseert anything more than that the total balance of disorder in nature is steadily on  the increase. In individual sections of the universe, or in definite material systems, the movement may  cvery well be towards a higher degree of order, which is made possible because an adequate compensation  occurs in some other systems.”

The notion of feedbacks gives us a new way of thinking about the game played between the novel and the author, in as much as the author keeps adding and subtracting from the novel, as well as that played between the reader and the novel, in that the reader keeps decoding the novel. But the question Musil was gnawing on was whether the novel as a system could accommodate the character as a point determined by the irreversible progress from order to disorder inherent in the other administrative systems within the social world that give the character a content.