Poincare Quote Regarding Instability

Question

Poincare wrote (english translation of the french) in 1903: ``A very small cause that escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.''

Where?* Poincare wrote a lot of small papers, and gave many talks, and traveled.

*Incidentally, Leibniz and Lichtenberg wrote similar things earlier, so this is a not a priority dispute question regarding instability. Later, but before modern citations, there was also, A. Compton, A. Lotka, E. Milne, J. Wheeler (a cigarette smoke causes a hurricane here and not elsewhere, about a year prior E. Lorenz bird and butterfly examples), etc, so definitely not a priority question.

Answer 1

This quote is taken from Poincaré's Science and Method, first pubished in 1908.

The linked webpage from archive.org is an English language translation.

The quote appears in section IV (titled Chance) of chapter I (titled The Scientist and Science). In the linked archive.org facsimile, the quote begins at the bottom of page 66 and continues onto the top of page 67. To view it, first click on the enlarge icon then use the horizontal scroll bar located at the bottom of the screen to advance to page 66. Alternatively, just search for "small cause", then click on the location pointer above the scroll bar at the bottom of the screen.

Science and Method is one of a series of lectures given by Poincaré before the Société de Psychologie in Paris (1903?) and published as Science and Hypothesis, The Value of Science, and Science and Method.

Answer 2

In there philosophy of chance and probability theory, there is a type of instability which does not depends to initial conditions and trajectories, called structural instability which referes to changes in dynamical models, laws and equations. There is a Mathematical définition of structural instability by Pontryagin and Andronov. See the Book of Philosophy of Complex Systems,

Static instability which is based on initial conditions.

Poincaré said chance events are such as arise from the action of causes in which small differences produce great differences in effect, it follows immediately that chance effects must be random

For instability he gives an example in his paper about chance,1908

Great differences in the cause and small differences in the effect Flammarion once imagined an observer moving away from the earth at a velocity greater than that of light. For him time would have its sign changed, history would be reversed, and Waterloo would come before Austerlitz. Well, for this observer effects and causes would be inverted, unstable equilibrium would no longer be the exception; on account of the universal irreversibility, everything would seem to him to come out of a kind of chaos in unstable equilibrium, and the whole of nature would appear to him to be given up to chance.