2 Feb 2010

The Infinitesimal Straw that Broke the Camel's Back. Prigogine & Stengers. Order out of Chaos / La Nouvelle Alliance

by Corry Shores
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[The following is quotation.]

The Infinitesimal Straw that Broke the Camel's Back


Prigogine & Stengers

Order out of Chaos: Man's New Dialogue with Nature
La Nouvelle Alliance: Métamorphose de la science

We have mentioned that trajectories correspond to deterministic laws; once an initial state is given, the dynamic laws of motion permit the calculation of trajectories at each point in the future or the past. However, a trajectory may become intrinsically indeterminate at certain singular points. For instance, a rigid pendulum may display two qualitatively different types of behavior - it may either oscillate or swing around its points of suspension. If the initial push is just enough to bring it into a vertical position with zero velocity, the direction in which it will fall, and therefore the nature of its motion, are indeterminate. An infinitesimal perturbation would be enough to set it rotating or oscillating. [...]

It is significant that Maxwell had already stressed the importance of these singular points. After describing the explosion of gun cotton, he goes to say:

In all such cases there is one common circumstance - the system has the quantity of potential energy, which is capable of being transformed into motion, but which cannot begin to be so transformed till the system has reached a certain configuration, to attain which requires an expenditure of work, which in certain cases may be infinitesimally small, and in general bears no definite proportion to the energy developed in consequence thereof. For example, the rock loosed by frost and balanced on a singular point of the mountain-side, the little spark which kindles the great forest, the little word which sets the world a fighting, the little scruple which prevents a man from doing his will, the little spore which blights all the potatoes, the little gemmule which makes us philosophers or idiots. Every existence above a certain rank has its singular points: the higher the rank, the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being, may produce results of the greatest importance. All great results produced by human endeavor depend on taking advantage of these singular states when they occur." [footnote 14. MAXWELL J.C., Science and Free Will, in CAMPBELL L. and GARNETT W., op. cit., p. 443. L. CAMPBELL & W. GARNETT , The Life of James Clerk Maxwell (London, Macmillan, 1882).] [73b.d/320c, emphasis mine]

Ainsi, des chercheurs notèrent qu'une trajectoire peut devenir intrinsèquement indéterminée en certains points singuliers. Un pendule rigide peut avoir deux types de comportements qualitativement différents : il peut soit osciller soit tourner autour de son point de suspension. Si son impulsion initiale est exactement suffisante pour le faire aboutir avec une vitesse nulle en position verticale, la direction vers laquelle il retombera, et donc la nature de son mouvement, est indéterminée : une perturbation infinitésimale suffit à entraîner soit une rotation soit une oscillation. [...]

Il est remarquable de constater que Maxwell avait déjà souligné l'importance de tels points singuliers : « Dans tous les cas de ce genre (Maxwell vient de décrire l'explosion du coton fulminant), il y a une circonstance commune : le système possède une quantité d'énergie potentielle qui peut être transformée en mouvement mais ne peut commencer à l'être que lorsque le système a atteint une certaine configuration, ce qui nécessite une dépense de travail, qui peut être infinitésimale et est en général sans commune mesure avec l'énergie qu'elle permet de libérer. Ainsi, le rocher détaché par le gel et en équilibre sur un point singulier du flanc de la montagne, la petite étincelle qui embrase l'immense forêt, le petit mot qui met le monde en guerre, le petit scrupule qui empêche l'homme de faire ce qu'il veut, le petit spore qui gâte toutes les pommes de terre, la petite gemmule qui fait de nous des philosophes ou des idiots. Chaque existence à partir d'un certain niveau a ses points singuliers : plus élevé le niveau, plus nombreux les points. En ces points, des influences, dont la taille physique est trop petite pour être prise en compte par un être fini, peuvent produire des résultats de la plus grande importance. Tous les grands résultats produits par les entreprises humaines dépendent de la manière dont on prend avantage de ces états singuliers, lorsqu'ils se présentent.» [note I. MAXWELL J.C., Science and Free Will, in CAMPBELL L. et GARNETT W., op. cit., p. 443] [85b-86]

[Also see the section on catastrophe theory towards the end of this entry.]

Prigogine, Ilya, and Isabelle Stengers. Order out of Chaos: Man's New Dialogue with Nature. London: Heinemann, 1984.

Prigogine, Ilya, and Isabelle Stengers. La Nouvelle Alliance: Métamorphose de la science. Paris: Éditions Gallimard, 1979.

1 comment:

  1. in some regard this is the basis of artistic creation and possibly social revolutions. a seemingly small amount of effort or work can set into motion a whole series of events if certain components in the system are in a particular arrangement which once engaged set off something way bigger than the initial effort itself. it also reinforces the case for constant experimentation in regards to our work. i recall the first time i wrote music in ink opposed to pencil. that small event set off a significant series of changes in the way i thought about music. that small experiment has radically altered my life. i could not have known. so it becomes of question of how do we identify these potential systems and then the needed infinitesimal perturbation that will set a new reality into motion.