Wednesday, March 22, 2006

Henri Poincaré’s “Three guises of chance”

Human action and contingency seem to go together in historical events.

If life is viewed as partially a game of chance like poker, let's say, involving both deterministic skill and indeterministic chance, then surviving for the next round and play at even higher stakes is a matter of chance (cf. Taleb's discussion of survivorship bias).

Henri Poincaré, the nineteenth century French mathematician, argued that chance has more than one form:

1. A statistically random phenomenon. [Noise]
2. The amplification of a microcause. [Sensitive dynamics]
3. A function of our analytical blindness. [Scientific method]

Beyerchen, in the paper linked to above, maps Poincare’s forms of chance to Clausewitz’s observations on war. Clausewitz holds that chance is one of the three important factors shaping war:

(a) “chance and probability, faced or generated by the commander and his army”; [addressed by Poincare and enumerated here]

(b) “the blind, natural force of violence, hatred, and enmity among the masses of people”; [the force driving high-intensity warfare such that when one society is faced with possible subordination and enslavement by another society, a large fraction of a society serving as individual soldiers lays down their lives in battle]

(c) “war's rational subordination to the policy of the government”
[some rational strategy given the constraints a society faces, like manpower accumulation in the context of low population density mainland Southeast Asian history]

Look at each of Poincare’s forms of chance and its relation to Clausewitz's observations:

1. A statistically random phenomenon:

i. Poincare this as “the familiar form of chance that can arise where permutations of small causes are extremely numerous or where the number of variables is quite large. This form of chance can be calculated by statistical methods. The very large number of interactions produces a disorganization sufficient to result in a symmetrical (i.e., Gaussian or bell curve) [Gaussian noise] probability distribution. Nothing significant is left of the initial conditions, and the history of the system no longer matters.”

ii. For Clausewitz, "in the whole range of human activities, war most closely resembles a game of cards." This analogy “suggests not only the ability to calculate probabilities, but knowledge of human psychology in ‘reading’ the other players, sensing when to take risks, and so on.”

iii. “Clausewitz certainly understands that the number of variables in war can be enormous, and that a rather special aptitude is needed to cope with the chance and complexity involved…The man responsible for evaluating the whole must bring to his task the quality of intuition that perceives the truth at every point. Otherwise a chaos of opinions and considerations would arise, and fatally entangle judgment.”

2. The amplification of a micro-cause:

i. Butterfly effects are small causes with large effects later on.

ii. For example, the flap of a butterfly’s wings in Brazil set off a tornado in Texas.

iii. Another example, the agency of single human historical actors in worlds where strong deterministic social and environmental constraints drive most action and history.

v. In mathematics the butterfly effect is called “sensitive dependence on initial conditions” in a system of differential equations (dynamical system).

vi. Small variations of the initial conditions of a dynamical system may produce large variations in the long term behavior of the system.

vii. An example from physics, ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position.

viii. Because of this, deterministic phenomena must be treated as stochastic processes.

ix. “Recurrence, the approximate return of a system towards its initial conditions, together with the sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range—approximately a week, in the case of weather.”

x. “It is not just, as earlier probabilists had said, that there are a great many causal variables, and we are ignorant of most of them; even in situations where there there are only a few variables, but arbitrarily small differences in the starting initial state can be rapidly amplified into large differences in the latter state, you'll often have no choice but to treat the dynamics as largely random.”. …This is explained very clearly in his essay "On Chance" in Science and Method.” (From:

xi. Poincare: “A very slight cause, which escapes us, determines a considerable effect which we can not help seeing, and then we say this effect is due to chance. If we could know exactly the laws of nature and the situation of the universe at the initial instant, we should be able to predict exactly the situation of this same universe at a subsequent instant. But even when the natural laws should have no further secret for us, we could know the initial situation only approximately. If that permits us to foresee the subsequent situation with the same degree of approximation, this is all we require, [and] we say the phenomenon has been predicted, that is ruled by laws. But this is not always the case; it may happen that slight differences in the initial conditions produce very great differences in the final phenomenon; a slight error in the former would make an enormous error in the latter. Prediction becomes impossible and we have the fortuitous phenomenon.”

3. A function of our analytical blindness:

i. The problems in creating foolproof scientific methods revealed by the philosophy of science.

ii. “…inability to see the universe as an interconnected whole.”

iii. “Our weakness forbids our considering the entire universe and makes us cut it up into slices. We try to do this as little artificially as possible. And yet it happens from time to time that two of these slices react upon each other. The effects of this mutual action then seem to us to be due to chance.”

iv. Clausewitz’s "diversity and indistinct boundary of all relationships"

v. “Efforts were therefore made to equip the conduct of war with principles, rules, or even systems. This did present a positive goal, but people failed to take an adequate account of the endless complexities involved. As we have seen, the conduct of war branches out in almost all directions and has no definite limits; while any system, any model, has the finite nature of a synthesis [in the sense of synthetic or man-made]. An irreconcilable conflict exists between this type of theory and actual practice....[These attempts] aim at fixed values; but in war everything is uncertain, and calculations have to be made with variable quantities. They direct the inquiry exclusively toward physical quantities, whereas all military action is entwined with psychological forces and effects. They consider only unilateral action, whereas war consists of continuous interaction of opposites.”

vi. “Decisive results can often rest on particular factors that are ‘details known only to those who were on the spot.’ Attempts to reconstruct cause and effect always face the lack of precise information:

“Nowhere in life is this so common as in war, where the facts are seldom fully known and the underlying motives even less so. They may be intentionally concealed by those in command, or, if they happen to be transitory and accidental, history may not have recorded them at all.”

“We can never recover the precise initial conditions even of known developments in past wars, much less developments in current wars distorted by the fog of uncertainty. Interactions at every scale within armies and between adversaries amplify microcauses and produce unexpected macroeffects. Since interaction is intrinsic to the nature of war, it cannot be eliminated. The precise knowledge needed to anticipate the effects of interaction is unattainable.”